| L(s) = 1 | − 2.31·2-s − 1.11·3-s + 3.35·4-s + 2.59·6-s + 1.28·7-s − 3.13·8-s − 1.74·9-s − 1.10·11-s − 3.75·12-s + 3.93·13-s − 2.97·14-s + 0.553·16-s + 2.17·17-s + 4.04·18-s − 3.54·19-s − 1.43·21-s + 2.54·22-s − 5.81·23-s + 3.51·24-s − 9.11·26-s + 5.31·27-s + 4.30·28-s − 4.44·29-s − 0.859·31-s + 4.99·32-s + 1.23·33-s − 5.03·34-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.646·3-s + 1.67·4-s + 1.05·6-s + 0.485·7-s − 1.11·8-s − 0.582·9-s − 0.331·11-s − 1.08·12-s + 1.09·13-s − 0.793·14-s + 0.138·16-s + 0.527·17-s + 0.952·18-s − 0.812·19-s − 0.313·21-s + 0.542·22-s − 1.21·23-s + 0.717·24-s − 1.78·26-s + 1.02·27-s + 0.814·28-s − 0.825·29-s − 0.154·31-s + 0.883·32-s + 0.214·33-s − 0.863·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4549318805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4549318805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 + 1.11T + 3T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 + 5.81T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 + 0.859T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 + 2.52T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 9.19T + 59T^{2} \) |
| 61 | \( 1 + 9.71T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 6.93T + 73T^{2} \) |
| 79 | \( 1 - 1.95T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.325032027148379968737821739835, −7.57131414055580569741358040698, −6.85049269184058909431041730049, −6.00773569080122393361905588936, −5.59554383733855377585730403875, −4.49736904420336767635287121192, −3.49372154307138709833449920114, −2.32468771349420276602188044106, −1.54085887549822288209433401996, −0.48214637460050279909613024913,
0.48214637460050279909613024913, 1.54085887549822288209433401996, 2.32468771349420276602188044106, 3.49372154307138709833449920114, 4.49736904420336767635287121192, 5.59554383733855377585730403875, 6.00773569080122393361905588936, 6.85049269184058909431041730049, 7.57131414055580569741358040698, 8.325032027148379968737821739835
