| L(s) = 1 | + 5.90e4·3-s − 8.90e6·5-s − 5.87e8·7-s + 3.48e9·9-s + 1.42e11·11-s + 1.49e11·13-s − 5.26e11·15-s + 9.95e12·17-s + 5.02e12·19-s − 3.47e13·21-s − 3.25e14·23-s − 3.97e14·25-s + 2.05e14·27-s + 1.07e15·29-s − 3.98e15·31-s + 8.40e15·33-s + 5.23e15·35-s − 3.35e16·37-s + 8.80e15·39-s + 4.54e16·41-s + 8.38e16·43-s − 3.10e16·45-s + 5.34e16·47-s − 2.13e17·49-s + 5.87e17·51-s + 1.14e18·53-s − 1.26e18·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.407·5-s − 0.786·7-s + 0.333·9-s + 1.65·11-s + 0.299·13-s − 0.235·15-s + 1.19·17-s + 0.188·19-s − 0.454·21-s − 1.63·23-s − 0.833·25-s + 0.192·27-s + 0.472·29-s − 0.873·31-s + 0.955·33-s + 0.320·35-s − 1.14·37-s + 0.173·39-s + 0.529·41-s + 0.591·43-s − 0.135·45-s + 0.148·47-s − 0.381·49-s + 0.691·51-s + 0.896·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.596468938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.596468938\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.90e4T \) |
| good | 5 | \( 1 + 8.90e6T + 4.76e14T^{2} \) |
| 7 | \( 1 + 5.87e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.42e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 1.49e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 9.95e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 5.02e12T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.25e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.07e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 3.98e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.35e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 4.54e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 8.38e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 5.34e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.14e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 5.97e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 4.12e17T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.42e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 7.25e16T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.97e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 5.78e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.48e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 5.56e19T + 8.65e40T^{2} \) |
| 97 | \( 1 + 8.39e20T + 5.27e41T^{2} \) |
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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
−11.69430295551515777004525640187, −10.08901326965293613514606621649, −9.251192180092360863362166898513, −8.099247992820229316802168536809, −6.92624292090448180832687494362, −5.82227097490134446351516159869, −3.96728517709648300096028138414, −3.47947767312057644094550367615, −1.92343713301076485436274408925, −0.72401769586909199512368334911,
0.72401769586909199512368334911, 1.92343713301076485436274408925, 3.47947767312057644094550367615, 3.96728517709648300096028138414, 5.82227097490134446351516159869, 6.92624292090448180832687494362, 8.099247992820229316802168536809, 9.251192180092360863362166898513, 10.08901326965293613514606621649, 11.69430295551515777004525640187
