L(s) = 1 | − 2.38·3-s − 5-s + 3.11·7-s + 2.67·9-s + 0.421·11-s + 3.75·13-s + 2.38·15-s − 5.64·17-s − 1.71·19-s − 7.41·21-s + 0.413·23-s + 25-s + 0.771·27-s + 9.21·29-s − 3.25·31-s − 1.00·33-s − 3.11·35-s − 7.17·37-s − 8.94·39-s + 6.11·41-s + 6.53·43-s − 2.67·45-s + 7.84·47-s + 2.67·49-s + 13.4·51-s + 7.69·53-s − 0.421·55-s + ⋯ |
L(s) = 1 | − 1.37·3-s − 0.447·5-s + 1.17·7-s + 0.892·9-s + 0.126·11-s + 1.04·13-s + 0.615·15-s − 1.36·17-s − 0.393·19-s − 1.61·21-s + 0.0863·23-s + 0.200·25-s + 0.148·27-s + 1.71·29-s − 0.583·31-s − 0.174·33-s − 0.525·35-s − 1.17·37-s − 1.43·39-s + 0.954·41-s + 0.995·43-s − 0.398·45-s + 1.14·47-s + 0.382·49-s + 1.88·51-s + 1.05·53-s − 0.0567·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.168790753\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.168790753\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 + 2.38T + 3T^{2} \) |
| 7 | \( 1 - 3.11T + 7T^{2} \) |
| 11 | \( 1 - 0.421T + 11T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 17 | \( 1 + 5.64T + 17T^{2} \) |
| 19 | \( 1 + 1.71T + 19T^{2} \) |
| 23 | \( 1 - 0.413T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 + 7.17T + 37T^{2} \) |
| 41 | \( 1 - 6.11T + 41T^{2} \) |
| 43 | \( 1 - 6.53T + 43T^{2} \) |
| 47 | \( 1 - 7.84T + 47T^{2} \) |
| 53 | \( 1 - 7.69T + 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 + 0.578T + 61T^{2} \) |
| 67 | \( 1 + 2.21T + 67T^{2} \) |
| 71 | \( 1 - 7.37T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 2.46T + 97T^{2} \) |
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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
−8.275388308143066250970534649696, −7.07497078420763667386796924970, −6.71832201905825144158862393079, −5.81862397962964466329289812768, −5.31092964105069692022066970488, −4.35881824785216870822108651795, −4.15406706930191278841311124666, −2.69332894033164703336877995767, −1.55660227918472178489867373125, −0.64703771349796756909075319573,
0.64703771349796756909075319573, 1.55660227918472178489867373125, 2.69332894033164703336877995767, 4.15406706930191278841311124666, 4.35881824785216870822108651795, 5.31092964105069692022066970488, 5.81862397962964466329289812768, 6.71832201905825144158862393079, 7.07497078420763667386796924970, 8.275388308143066250970534649696