L(s) = 1 | + (0.923 + 0.382i)2-s + (0.541 − 0.541i)3-s + (0.707 + 0.707i)4-s + (0.707 − 0.292i)6-s + (0.382 + 0.923i)8-s + 0.414i·9-s − 1.41i·11-s + 0.765·12-s + (−0.541 − 0.541i)13-s + i·16-s + (−0.158 + 0.382i)18-s + 19-s + (0.541 − 1.30i)22-s + (0.707 + 0.292i)24-s + (−0.292 − 0.707i)26-s + (0.765 + 0.765i)27-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)2-s + (0.541 − 0.541i)3-s + (0.707 + 0.707i)4-s + (0.707 − 0.292i)6-s + (0.382 + 0.923i)8-s + 0.414i·9-s − 1.41i·11-s + 0.765·12-s + (−0.541 − 0.541i)13-s + i·16-s + (−0.158 + 0.382i)18-s + 19-s + (0.541 − 1.30i)22-s + (0.707 + 0.292i)24-s + (−0.292 − 0.707i)26-s + (0.765 + 0.765i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.365630768\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.365630768\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
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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
−9.177496640036474792161557210132, −8.262211297894012234865409372782, −7.86551993188857415018544727521, −7.08528574953988993531879531992, −6.21569676264673889515586394503, −5.39080322661071122592310584816, −4.70277394112873954030995965499, −3.27265439433177827257583767795, −2.97423700743361914369795117728, −1.64253306514119950758168321757,
1.65932673570306424922656240100, 2.68297613835421237226703120875, 3.59628911183967796773882820332, 4.39820891415092260259942552080, 5.02130142531765024694614355813, 6.05368929955030553896256930501, 7.05121375546338107161080698818, 7.50172271964546948309979551305, 8.915385147966603273723071565371, 9.549056488400845637956126800767