| L(s) = 1 | − 3-s − 2·9-s − 11-s − 7·17-s + 4·19-s + 9·25-s + 5·27-s + 33-s − 9·41-s − 6·43-s + 8·49-s + 7·51-s − 4·57-s + 13·59-s − 22·67-s − 8·73-s − 9·75-s + 81-s + 18·83-s + 5·89-s − 10·97-s + 2·99-s − 4·107-s − 17·113-s − 121-s + 9·123-s + 127-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.301·11-s − 1.69·17-s + 0.917·19-s + 9/5·25-s + 0.962·27-s + 0.174·33-s − 1.40·41-s − 0.914·43-s + 8/7·49-s + 0.980·51-s − 0.529·57-s + 1.69·59-s − 2.68·67-s − 0.936·73-s − 1.03·75-s + 1/9·81-s + 1.97·83-s + 0.529·89-s − 1.01·97-s + 0.201·99-s − 0.386·107-s − 1.59·113-s − 0.0909·121-s + 0.811·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.487124882825142272220401033767, −7.975145533326363299500184212411, −7.27356344285136560812155298853, −6.95745442372012316684792408307, −6.51265586658143150507565159324, −6.10523578291370215974885281386, −5.39226580230390723405127601316, −5.12481708974094531168754867854, −4.66141917895406151612324732439, −4.05230306442650794396250264101, −3.24495655547011910756532013046, −2.81618069839981920282744814383, −2.13145787359301351228846848803, −1.11543574847005501664797620421, 0,
1.11543574847005501664797620421, 2.13145787359301351228846848803, 2.81618069839981920282744814383, 3.24495655547011910756532013046, 4.05230306442650794396250264101, 4.66141917895406151612324732439, 5.12481708974094531168754867854, 5.39226580230390723405127601316, 6.10523578291370215974885281386, 6.51265586658143150507565159324, 6.95745442372012316684792408307, 7.27356344285136560812155298853, 7.975145533326363299500184212411, 8.487124882825142272220401033767
