| L(s) = 1 | + 3-s − 2·4-s + 5-s + 7-s − 2·9-s + 3·11-s − 2·12-s − 5·13-s + 15-s + 4·16-s + 3·17-s − 2·19-s − 2·20-s + 21-s + 6·23-s + 25-s − 5·27-s − 2·28-s − 3·29-s + 4·31-s + 3·33-s + 35-s + 4·36-s + 2·37-s − 5·39-s + 12·41-s + 10·43-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s + 0.727·17-s − 0.458·19-s − 0.447·20-s + 0.218·21-s + 1.25·23-s + 1/5·25-s − 0.962·27-s − 0.377·28-s − 0.557·29-s + 0.718·31-s + 0.522·33-s + 0.169·35-s + 2/3·36-s + 0.328·37-s − 0.800·39-s + 1.87·41-s + 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.894748881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.894748881\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 47 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.14903506400788, −13.65808613708156, −13.10036819282079, −12.57869943230203, −12.21100072338902, −11.56324467134773, −11.03934615279457, −10.41607678691302, −9.758756457102174, −9.465426109532644, −8.961331924744946, −8.657177493926324, −7.932205738893120, −7.527347396250062, −6.965048783006685, −6.122391304952919, −5.590517987923631, −5.189757953955260, −4.466985612093940, −4.051209707319724, −3.349279849093354, −2.613918311366194, −2.235700998928465, −1.145498296175759, −0.6177865421306052,
0.6177865421306052, 1.145498296175759, 2.235700998928465, 2.613918311366194, 3.349279849093354, 4.051209707319724, 4.466985612093940, 5.189757953955260, 5.590517987923631, 6.122391304952919, 6.965048783006685, 7.527347396250062, 7.932205738893120, 8.657177493926324, 8.961331924744946, 9.465426109532644, 9.758756457102174, 10.41607678691302, 11.03934615279457, 11.56324467134773, 12.21100072338902, 12.57869943230203, 13.10036819282079, 13.65808613708156, 14.14903506400788
