| L(s) = 1 | − 3.88e5·3-s − 1.05e8·5-s + 3.81e9·7-s + 5.67e10·9-s − 2.52e11·11-s + 3.59e12·13-s + 4.08e13·15-s + 2.34e14·17-s + 6.23e14·19-s − 1.48e15·21-s − 3.58e15·23-s − 8.82e14·25-s + 1.45e16·27-s + 2.05e16·29-s + 1.36e17·31-s + 9.79e16·33-s − 4.00e17·35-s + 1.23e18·37-s − 1.39e18·39-s + 1.40e18·41-s − 2.18e17·43-s − 5.96e18·45-s − 8.67e18·47-s − 1.28e19·49-s − 9.09e19·51-s + 7.63e19·53-s + 2.64e19·55-s + ⋯ |
| L(s) = 1 | − 1.26·3-s − 0.962·5-s + 0.728·7-s + 0.602·9-s − 0.266·11-s + 0.555·13-s + 1.21·15-s + 1.65·17-s + 1.22·19-s − 0.922·21-s − 0.785·23-s − 0.0740·25-s + 0.502·27-s + 0.313·29-s + 0.963·31-s + 0.337·33-s − 0.701·35-s + 1.14·37-s − 0.703·39-s + 0.397·41-s − 0.0359·43-s − 0.580·45-s − 0.512·47-s − 0.469·49-s − 2.09·51-s + 1.13·53-s + 0.256·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(1.287608097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287608097\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 3.88e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.05e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 3.81e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 2.52e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 3.59e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 2.34e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.23e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 3.58e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 2.05e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.36e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.23e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 1.40e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 2.18e17T + 3.71e37T^{2} \) |
| 47 | \( 1 + 8.67e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 7.63e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.01e18T + 5.36e40T^{2} \) |
| 61 | \( 1 + 2.87e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.47e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 7.64e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.49e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 1.02e22T + 4.42e43T^{2} \) |
| 83 | \( 1 - 7.71e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 4.58e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 1.13e23T + 4.96e45T^{2} \) |
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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
−10.93314286502785279700492270371, −9.879913566658290617061240129422, −8.181382872519304270615515420223, −7.53467782161902787436728518598, −6.10060066659180529612633697685, −5.27736610147200845641926949336, −4.27940824675953138290896803481, −3.09079121385167159095496484487, −1.30376338176067444772754165917, −0.57732197670558260913795131502,
0.57732197670558260913795131502, 1.30376338176067444772754165917, 3.09079121385167159095496484487, 4.27940824675953138290896803481, 5.27736610147200845641926949336, 6.10060066659180529612633697685, 7.53467782161902787436728518598, 8.181382872519304270615515420223, 9.879913566658290617061240129422, 10.93314286502785279700492270371
