| L(s) = 1 | − 5.33·2-s + 20.4·4-s + 16.4·5-s + 9.67·7-s − 66.1·8-s − 87.4·10-s − 27.5·11-s − 51.5·14-s + 189.·16-s − 107.·17-s − 2.24·19-s + 335.·20-s + 147.·22-s − 41.8·23-s + 144.·25-s + 197.·28-s − 61.6·29-s + 191.·31-s − 480.·32-s + 575.·34-s + 158.·35-s + 98.4·37-s + 11.9·38-s − 1.08e3·40-s + 30.7·41-s + 238.·43-s − 563.·44-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 2.55·4-s + 1.46·5-s + 0.522·7-s − 2.92·8-s − 2.76·10-s − 0.756·11-s − 0.984·14-s + 2.95·16-s − 1.53·17-s − 0.0271·19-s + 3.74·20-s + 1.42·22-s − 0.379·23-s + 1.15·25-s + 1.33·28-s − 0.394·29-s + 1.11·31-s − 2.65·32-s + 2.90·34-s + 0.767·35-s + 0.437·37-s + 0.0511·38-s − 4.29·40-s + 0.117·41-s + 0.845·43-s − 1.92·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.33T + 8T^{2} \) |
| 5 | \( 1 - 16.4T + 125T^{2} \) |
| 7 | \( 1 - 9.67T + 343T^{2} \) |
| 11 | \( 1 + 27.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.24T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 61.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 98.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 30.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 511.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 484.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 444.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 190.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 484.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 957.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 375.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 715.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 65.5T + 9.12e5T^{2} \) |
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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
−8.925371490925232348903664178095, −8.070712424255959522490523091229, −7.35921782573344812896806522671, −6.36765556057263244315687460487, −5.89655702028838734401575589108, −4.67342567910907310653353159670, −2.72585200419849673002162077234, −2.14227727077294197855542567236, −1.30028219501255062181861427574, 0,
1.30028219501255062181861427574, 2.14227727077294197855542567236, 2.72585200419849673002162077234, 4.67342567910907310653353159670, 5.89655702028838734401575589108, 6.36765556057263244315687460487, 7.35921782573344812896806522671, 8.070712424255959522490523091229, 8.925371490925232348903664178095
