| L(s) = 1 | + 31.7·2-s + 81·3-s + 493.·4-s − 2.52e3·5-s + 2.56e3·6-s − 581.·8-s + 6.56e3·9-s − 8.01e4·10-s − 5.31e4·11-s + 3.99e4·12-s + 1.65e5·13-s − 2.04e5·15-s − 2.71e5·16-s + 1.76e5·17-s + 2.08e5·18-s + 9.48e5·19-s − 1.24e6·20-s − 1.68e6·22-s + 1.54e6·23-s − 4.71e4·24-s + 4.44e6·25-s + 5.25e6·26-s + 5.31e5·27-s − 1.17e6·29-s − 6.49e6·30-s + 9.44e4·31-s − 8.30e6·32-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.577·3-s + 0.964·4-s − 1.80·5-s + 0.809·6-s − 0.0502·8-s + 0.333·9-s − 2.53·10-s − 1.09·11-s + 0.556·12-s + 1.60·13-s − 1.04·15-s − 1.03·16-s + 0.513·17-s + 0.467·18-s + 1.66·19-s − 1.74·20-s − 1.53·22-s + 1.14·23-s − 0.0289·24-s + 2.27·25-s + 2.25·26-s + 0.192·27-s − 0.308·29-s − 1.46·30-s + 0.0183·31-s − 1.39·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.043335682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.043335682\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 31.7T + 512T^{2} \) |
| 5 | \( 1 + 2.52e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 5.31e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.65e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.76e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.48e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.54e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.17e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.44e4T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.25e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.92e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.17e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.43e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.55e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.61e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.72e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.40e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.76e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.45e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.51e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.67e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.47e9T + 7.60e17T^{2} \) |
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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
−11.55899864112545796534749121745, −10.85313381394893297998697326568, −9.029740077148602031207899172044, −7.968378333048634646031457274059, −7.17543400743001811237186680459, −5.61072366822954742270182662938, −4.50032646659278045781810067183, −3.50450446938047568216376311696, −3.00019817465117586483288399569, −0.824515682054090357519419660213,
0.824515682054090357519419660213, 3.00019817465117586483288399569, 3.50450446938047568216376311696, 4.50032646659278045781810067183, 5.61072366822954742270182662938, 7.17543400743001811237186680459, 7.968378333048634646031457274059, 9.029740077148602031207899172044, 10.85313381394893297998697326568, 11.55899864112545796534749121745
