| L(s) = 1 | + 93·3-s + 125·5-s − 343·7-s + 6.46e3·9-s + 2.16e3·11-s − 1.66e3·13-s + 1.16e4·15-s − 3.57e4·17-s − 2.02e4·19-s − 3.18e4·21-s + 4.21e4·23-s + 1.56e4·25-s + 3.97e5·27-s − 1.11e5·29-s + 2.69e5·31-s + 2.01e5·33-s − 4.28e4·35-s + 5.32e5·37-s − 1.54e5·39-s + 1.58e5·41-s + 5.21e5·43-s + 8.07e5·45-s + 9.39e5·47-s + 1.17e5·49-s − 3.32e6·51-s − 4.08e5·53-s + 2.70e5·55-s + ⋯ |
| L(s) = 1 | + 1.98·3-s + 0.447·5-s − 0.377·7-s + 2.95·9-s + 0.490·11-s − 0.209·13-s + 0.889·15-s − 1.76·17-s − 0.676·19-s − 0.751·21-s + 0.722·23-s + 1/5·25-s + 3.88·27-s − 0.851·29-s + 1.62·31-s + 0.976·33-s − 0.169·35-s + 1.72·37-s − 0.416·39-s + 0.358·41-s + 1.00·43-s + 1.32·45-s + 1.32·47-s + 1/7·49-s − 3.51·51-s − 0.376·53-s + 0.219·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.023461239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.023461239\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p^{3} T \) |
| 7 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 - 31 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 197 p T + p^{7} T^{2} \) |
| 13 | \( 1 + 1661 T + p^{7} T^{2} \) |
| 17 | \( 1 + 35771 T + p^{7} T^{2} \) |
| 19 | \( 1 + 20222 T + p^{7} T^{2} \) |
| 23 | \( 1 - 42130 T + p^{7} T^{2} \) |
| 29 | \( 1 + 111789 T + p^{7} T^{2} \) |
| 31 | \( 1 - 269504 T + p^{7} T^{2} \) |
| 37 | \( 1 - 532774 T + p^{7} T^{2} \) |
| 41 | \( 1 - 158056 T + p^{7} T^{2} \) |
| 43 | \( 1 - 521874 T + p^{7} T^{2} \) |
| 47 | \( 1 - 939733 T + p^{7} T^{2} \) |
| 53 | \( 1 + 408384 T + p^{7} T^{2} \) |
| 59 | \( 1 - 522172 T + p^{7} T^{2} \) |
| 61 | \( 1 - 350080 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3931176 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1194016 T + p^{7} T^{2} \) |
| 73 | \( 1 - 998350 T + p^{7} T^{2} \) |
| 79 | \( 1 - 2120709 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1746708 T + p^{7} T^{2} \) |
| 89 | \( 1 + 10077740 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6238295 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.310856142700045964390073704600, −8.938784345521179340593204862736, −8.047947384715294617251081455186, −7.06989129710357733395268937244, −6.34828391652912194456009179766, −4.55782253145449557394573063654, −3.91989989836331594225487743865, −2.65823716486276100911316390675, −2.23930582823280443556232223492, −0.984664456895481641452636411850,
0.984664456895481641452636411850, 2.23930582823280443556232223492, 2.65823716486276100911316390675, 3.91989989836331594225487743865, 4.55782253145449557394573063654, 6.34828391652912194456009179766, 7.06989129710357733395268937244, 8.047947384715294617251081455186, 8.938784345521179340593204862736, 9.310856142700045964390073704600
