| L(s) = 1 | + 485·9-s − 906·11-s − 1.02e3·16-s + 3.10e3·19-s + 9.97e3·29-s + 2.38e3·31-s − 3.45e3·41-s − 2.40e3·49-s + 9.16e3·59-s − 2.49e4·61-s + 1.03e5·71-s − 1.25e5·79-s + 1.76e5·81-s + 2.94e5·89-s − 4.39e5·99-s − 2.37e4·101-s − 2.18e5·109-s + 2.93e5·121-s + 127-s + 131-s + 137-s + 139-s − 4.96e5·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1.99·9-s − 2.25·11-s − 16-s + 1.97·19-s + 2.20·29-s + 0.445·31-s − 0.321·41-s − 1/7·49-s + 0.342·59-s − 0.859·61-s + 2.43·71-s − 2.26·79-s + 2.98·81-s + 3.94·89-s − 4.50·99-s − 0.231·101-s − 1.76·109-s + 1.82·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 1.99·144-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.796783770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.796783770\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{3} T + p^{5} T^{2} )( 1 + p^{3} T + p^{5} T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 485 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 453 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 196375 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 159945 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 1550 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10136970 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4985 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 1192 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 17291590 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 1728 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 177074290 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 229690155 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 163714890 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4580 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12488 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2449091110 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 51792 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4122659470 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 62765 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7319042550 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 147300 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17105074865 T^{2} + p^{10} T^{4} \) |
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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
−12.06735326588730341561040450758, −11.73568312719317848069050645179, −10.81416398259059334432302750730, −10.59753810391452638004688101755, −9.949361732436961955761367947815, −9.847386884642949521396264582937, −9.151686362148485108103715834549, −8.381313126003868411913195068632, −7.74151209681761902256821820158, −7.58292236807515149963801734584, −6.87149161634215483327583783636, −6.45815906732561642241387220150, −5.46025930056446819366695991473, −4.79492160768369488594983183889, −4.75251036030282506388032019195, −3.70362858934648110183946782235, −2.87743572919821925751961825977, −2.30001645617406793007499947497, −1.29807473905808848987781302870, −0.58060314810936465986601744244,
0.58060314810936465986601744244, 1.29807473905808848987781302870, 2.30001645617406793007499947497, 2.87743572919821925751961825977, 3.70362858934648110183946782235, 4.75251036030282506388032019195, 4.79492160768369488594983183889, 5.46025930056446819366695991473, 6.45815906732561642241387220150, 6.87149161634215483327583783636, 7.58292236807515149963801734584, 7.74151209681761902256821820158, 8.381313126003868411913195068632, 9.151686362148485108103715834549, 9.847386884642949521396264582937, 9.949361732436961955761367947815, 10.59753810391452638004688101755, 10.81416398259059334432302750730, 11.73568312719317848069050645179, 12.06735326588730341561040450758
