| L(s) = 1 | − 0.347·2-s − 0.879·4-s + 0.347·5-s − 1.53·7-s + 0.652·8-s + 9-s − 0.120·10-s + 0.532·14-s + 0.652·16-s − 0.347·18-s − 0.347·19-s − 0.305·20-s − 0.879·25-s + 1.34·28-s + 31-s − 0.879·32-s − 0.532·35-s − 0.879·36-s + 0.120·38-s + 0.226·40-s + 1.87·41-s + 0.347·45-s − 47-s + 1.34·49-s + 0.305·50-s − 56-s + 1.53·59-s + ⋯ |
| L(s) = 1 | − 0.347·2-s − 0.879·4-s + 0.347·5-s − 1.53·7-s + 0.652·8-s + 9-s − 0.120·10-s + 0.532·14-s + 0.652·16-s − 0.347·18-s − 0.347·19-s − 0.305·20-s − 0.879·25-s + 1.34·28-s + 31-s − 0.879·32-s − 0.532·35-s − 0.879·36-s + 0.120·38-s + 0.226·40-s + 1.87·41-s + 0.347·45-s − 47-s + 1.34·49-s + 0.305·50-s − 56-s + 1.53·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7770061440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7770061440\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.347T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.347T + T^{2} \) |
| 7 | \( 1 + 1.53T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.347T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.87T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.53T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - 1.53T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.53T + T^{2} \) |
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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
−8.906877362359874053300948487293, −7.981554275738434380808690608279, −7.27803332687698477188027938461, −6.44008177226272622241026222712, −5.86195235124405501593723572798, −4.80101782074305773869212445175, −4.06293325768557475756715389773, −3.34643022340595556409566959721, −2.15127931846108192597560732419, −0.805259854826387142198382045383,
0.805259854826387142198382045383, 2.15127931846108192597560732419, 3.34643022340595556409566959721, 4.06293325768557475756715389773, 4.80101782074305773869212445175, 5.86195235124405501593723572798, 6.44008177226272622241026222712, 7.27803332687698477188027938461, 7.981554275738434380808690608279, 8.906877362359874053300948487293
