| L(s) = 1 | + 2.55·3-s + 2.41·5-s + 2.90·7-s + 3.53·9-s + 1.51·11-s − 1.98·13-s + 6.16·15-s + 5.83·17-s − 1.65·19-s + 7.41·21-s + 4.32·23-s + 0.825·25-s + 1.35·27-s + 2.50·29-s − 2.24·31-s + 3.88·33-s + 7.00·35-s − 2.57·37-s − 5.07·39-s − 3.53·41-s + 11.4·43-s + 8.52·45-s − 10.4·47-s + 1.41·49-s + 14.9·51-s + 0.504·53-s + 3.66·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 1.07·5-s + 1.09·7-s + 1.17·9-s + 0.458·11-s − 0.550·13-s + 1.59·15-s + 1.41·17-s − 0.379·19-s + 1.61·21-s + 0.902·23-s + 0.165·25-s + 0.261·27-s + 0.464·29-s − 0.404·31-s + 0.676·33-s + 1.18·35-s − 0.423·37-s − 0.812·39-s − 0.552·41-s + 1.74·43-s + 1.27·45-s − 1.52·47-s + 0.202·49-s + 2.08·51-s + 0.0692·53-s + 0.494·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.313174468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.313174468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 2.55T + 3T^{2} \) |
| 5 | \( 1 - 2.41T + 5T^{2} \) |
| 7 | \( 1 - 2.90T + 7T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 + 1.98T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 - 4.32T + 23T^{2} \) |
| 29 | \( 1 - 2.50T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + 2.57T + 37T^{2} \) |
| 41 | \( 1 + 3.53T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 0.504T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 + 6.97T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 - 7.20T + 71T^{2} \) |
| 73 | \( 1 - 9.90T + 73T^{2} \) |
| 79 | \( 1 + 9.15T + 79T^{2} \) |
| 83 | \( 1 + 4.01T + 83T^{2} \) |
| 89 | \( 1 + 0.235T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.117103670191193120428224408568, −7.59503558922832923229139440832, −6.84935552580628822898538856421, −5.86842017956089983189304646665, −5.16555183415092310751533310809, −4.40873492195571849462650443177, −3.43348867462744032454534162785, −2.71940467430669313392925953023, −1.88895019559789001370079647014, −1.31089545587314372062256819212,
1.31089545587314372062256819212, 1.88895019559789001370079647014, 2.71940467430669313392925953023, 3.43348867462744032454534162785, 4.40873492195571849462650443177, 5.16555183415092310751533310809, 5.86842017956089983189304646665, 6.84935552580628822898538856421, 7.59503558922832923229139440832, 8.117103670191193120428224408568
