| L(s) = 1 | + 6.25e3·25-s + 6.74e3·29-s + 6.24e4·81-s − 5.97e5·89-s + 5.50e5·101-s − 6.44e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.48e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 2·25-s + 1.48·29-s + 1.05·81-s − 7.99·89-s + 5.36·101-s − 4·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 4·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s + 1.28e−6·227-s + 1.26e−6·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.841986183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.841986183\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 38 T + 722 T^{2} - 38 p^{5} T^{3} + p^{10} T^{4} )( 1 + 38 T + 722 T^{2} + 38 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 366 T + 66978 T^{2} - 366 p^{5} T^{3} + p^{10} T^{4} )( 1 + 366 T + 66978 T^{2} + 366 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4838 T + 11703122 T^{2} - 4838 p^{5} T^{3} + p^{10} T^{4} )( 1 + 4838 T + 11703122 T^{2} + 4838 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - 1686 T + p^{5} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 211028098 T^{2} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 11862 T + 70353522 T^{2} - 11862 p^{5} T^{3} + p^{10} T^{4} )( 1 + 11862 T + 70353522 T^{2} + 11862 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 33334 T + 555577778 T^{2} - 33334 p^{5} T^{3} + p^{10} T^{4} )( 1 + 33334 T + 555577778 T^{2} + 33334 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 1041591898 T^{2} + p^{10} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 100434 T + 5043494178 T^{2} - 100434 p^{5} T^{3} + p^{10} T^{4} )( 1 + 100434 T + 5043494178 T^{2} + 100434 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 163262 T + 13327240322 T^{2} - 163262 p^{5} T^{3} + p^{10} T^{4} )( 1 + 163262 T + 13327240322 T^{2} + 163262 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + 149286 T + p^{5} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.604878053939352309160480905059, −8.350610918747401579927395183499, −7.919473860570151954798060201846, −7.76544271861674465070080492966, −7.23906622226802766926801772630, −7.15270626514867665391932791562, −6.72424467515651859389167455087, −6.64731529857238299211028350062, −6.28167928458418066693308458609, −5.86253934615149057804375964543, −5.67421104945410935181199131638, −5.17395895546955066559196490238, −4.99148362711286065466176611277, −4.67173215568852360500374362356, −4.36973444194392990515874687005, −3.84518059304789371438654971787, −3.79882461798327714459295445937, −2.91394974735310717036899983207, −2.83434547092922177862392860446, −2.81621098361900734285320379541, −1.98049146539260397135605988442, −1.61535627154551563979352343050, −1.10020759762390608823678768423, −0.819063421390769069647615944596, −0.27472225457193389611878470890,
0.27472225457193389611878470890, 0.819063421390769069647615944596, 1.10020759762390608823678768423, 1.61535627154551563979352343050, 1.98049146539260397135605988442, 2.81621098361900734285320379541, 2.83434547092922177862392860446, 2.91394974735310717036899983207, 3.79882461798327714459295445937, 3.84518059304789371438654971787, 4.36973444194392990515874687005, 4.67173215568852360500374362356, 4.99148362711286065466176611277, 5.17395895546955066559196490238, 5.67421104945410935181199131638, 5.86253934615149057804375964543, 6.28167928458418066693308458609, 6.64731529857238299211028350062, 6.72424467515651859389167455087, 7.15270626514867665391932791562, 7.23906622226802766926801772630, 7.76544271861674465070080492966, 7.919473860570151954798060201846, 8.350610918747401579927395183499, 8.604878053939352309160480905059
