| L(s) = 1 | − 3.04·3-s + 5-s − 0.574·7-s + 6.26·9-s − 2.57·11-s + 0.468·13-s − 3.04·15-s − 4.08·17-s − 19-s + 1.74·21-s + 1.51·23-s + 25-s − 9.92·27-s + 4.08·29-s − 9.92·31-s + 7.83·33-s − 0.574·35-s + 8.30·37-s − 1.42·39-s − 1.83·41-s + 0.574·43-s + 6.26·45-s + 7.09·47-s − 6.66·49-s + 12.4·51-s − 4.30·53-s − 2.57·55-s + ⋯ |
| L(s) = 1 | − 1.75·3-s + 0.447·5-s − 0.217·7-s + 2.08·9-s − 0.776·11-s + 0.129·13-s − 0.785·15-s − 0.991·17-s − 0.229·19-s + 0.381·21-s + 0.315·23-s + 0.200·25-s − 1.90·27-s + 0.758·29-s − 1.78·31-s + 1.36·33-s − 0.0971·35-s + 1.36·37-s − 0.228·39-s − 0.286·41-s + 0.0876·43-s + 0.933·45-s + 1.03·47-s − 0.952·49-s + 1.74·51-s − 0.591·53-s − 0.347·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6797264413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6797264413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3.04T + 3T^{2} \) |
| 7 | \( 1 + 0.574T + 7T^{2} \) |
| 11 | \( 1 + 2.57T + 11T^{2} \) |
| 13 | \( 1 - 0.468T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 - 4.08T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 - 8.30T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 - 0.574T + 43T^{2} \) |
| 47 | \( 1 - 7.09T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 - 2.68T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−7.84231054032249487494196981777, −7.16200753104689810033598514594, −6.38230218125881667655937325968, −6.05313070481054901167984689687, −5.19742009004109175099828235826, −4.77181309555998268359759433099, −3.89071484529974195704010412248, −2.64954969573472993865662319595, −1.62788027394006441054431806010, −0.47643738892083634241554504790,
0.47643738892083634241554504790, 1.62788027394006441054431806010, 2.64954969573472993865662319595, 3.89071484529974195704010412248, 4.77181309555998268359759433099, 5.19742009004109175099828235826, 6.05313070481054901167984689687, 6.38230218125881667655937325968, 7.16200753104689810033598514594, 7.84231054032249487494196981777
