| L(s) = 1 | − 9·3-s + 84.9·5-s − 141.·7-s + 81·9-s − 576.·11-s + 357.·13-s − 764.·15-s − 373.·17-s + 265.·19-s + 1.27e3·21-s − 529·23-s + 4.08e3·25-s − 729·27-s − 8.15e3·29-s − 607.·31-s + 5.19e3·33-s − 1.20e4·35-s − 1.07e3·37-s − 3.21e3·39-s − 1.27e4·41-s + 1.98e4·43-s + 6.87e3·45-s + 2.20e3·47-s + 3.21e3·49-s + 3.36e3·51-s + 1.11e4·53-s − 4.89e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·5-s − 1.09·7-s + 0.333·9-s − 1.43·11-s + 0.586·13-s − 0.877·15-s − 0.313·17-s + 0.168·19-s + 0.630·21-s − 0.208·23-s + 1.30·25-s − 0.192·27-s − 1.80·29-s − 0.113·31-s + 0.829·33-s − 1.65·35-s − 0.128·37-s − 0.338·39-s − 1.18·41-s + 1.63·43-s + 0.506·45-s + 0.145·47-s + 0.191·49-s + 0.181·51-s + 0.547·53-s − 2.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.491733332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.491733332\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 - 84.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 141.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 576.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 357.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 373.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 265.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 8.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 607.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.98e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.20e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.73e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.83e3T + 8.58e9T^{2} \) |
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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
−9.472277335902995454203497963875, −8.403390957292832853564081793166, −7.22417652730042978620219922233, −6.45187969985177788938675800835, −5.63757095100522399411690361062, −5.30124497030338370658222891993, −3.82881321969992512791566894329, −2.68129915454571853170867460507, −1.85147719716122185925649010832, −0.51814682351652762800446168732,
0.51814682351652762800446168732, 1.85147719716122185925649010832, 2.68129915454571853170867460507, 3.82881321969992512791566894329, 5.30124497030338370658222891993, 5.63757095100522399411690361062, 6.45187969985177788938675800835, 7.22417652730042978620219922233, 8.403390957292832853564081793166, 9.472277335902995454203497963875
