| L(s) = 1 | + 3.90e14·2-s + 2.16e22·3-s + 1.12e29·4-s + 1.39e33·5-s + 8.46e36·6-s + 1.75e40·7-s + 2.84e43·8-s − 1.65e45·9-s + 5.45e47·10-s − 5.41e48·11-s + 2.44e51·12-s + 1.04e52·13-s + 6.84e54·14-s + 3.03e55·15-s + 6.65e57·16-s − 2.56e58·17-s − 6.43e59·18-s − 4.34e60·19-s + 1.57e62·20-s + 3.80e62·21-s − 2.11e63·22-s − 7.48e64·23-s + 6.17e65·24-s − 5.66e65·25-s + 4.09e66·26-s − 8.18e67·27-s + 1.97e69·28-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 0.471·3-s + 2.84·4-s + 0.880·5-s + 0.923·6-s + 1.26·7-s + 3.61·8-s − 0.778·9-s + 1.72·10-s − 0.185·11-s + 1.33·12-s + 0.128·13-s + 2.47·14-s + 0.414·15-s + 4.23·16-s − 0.916·17-s − 1.52·18-s − 0.788·19-s + 2.50·20-s + 0.595·21-s − 0.362·22-s − 1.55·23-s + 1.70·24-s − 0.224·25-s + 0.251·26-s − 0.837·27-s + 3.59·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(96-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+95/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(48)\) |
\(\approx\) |
\(11.20473758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.20473758\) |
| \(L(\frac{97}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 3.90e14T + 3.96e28T^{2} \) |
| 3 | \( 1 - 2.16e22T + 2.12e45T^{2} \) |
| 5 | \( 1 - 1.39e33T + 2.52e66T^{2} \) |
| 7 | \( 1 - 1.75e40T + 1.92e80T^{2} \) |
| 11 | \( 1 + 5.41e48T + 8.55e98T^{2} \) |
| 13 | \( 1 - 1.04e52T + 6.67e105T^{2} \) |
| 17 | \( 1 + 2.56e58T + 7.80e116T^{2} \) |
| 19 | \( 1 + 4.34e60T + 3.03e121T^{2} \) |
| 23 | \( 1 + 7.48e64T + 2.31e129T^{2} \) |
| 29 | \( 1 - 2.03e69T + 8.46e138T^{2} \) |
| 31 | \( 1 - 3.72e70T + 4.77e141T^{2} \) |
| 37 | \( 1 + 1.43e74T + 9.53e148T^{2} \) |
| 41 | \( 1 + 4.16e76T + 1.63e153T^{2} \) |
| 43 | \( 1 - 4.94e76T + 1.51e155T^{2} \) |
| 47 | \( 1 - 4.71e79T + 7.06e158T^{2} \) |
| 53 | \( 1 - 9.21e80T + 6.40e163T^{2} \) |
| 59 | \( 1 + 1.44e84T + 1.70e168T^{2} \) |
| 61 | \( 1 - 2.68e84T + 4.03e169T^{2} \) |
| 67 | \( 1 - 3.24e86T + 2.99e173T^{2} \) |
| 71 | \( 1 - 2.78e87T + 7.40e175T^{2} \) |
| 73 | \( 1 - 2.94e88T + 1.03e177T^{2} \) |
| 79 | \( 1 - 9.20e89T + 1.88e180T^{2} \) |
| 83 | \( 1 + 2.26e90T + 2.05e182T^{2} \) |
| 89 | \( 1 - 3.95e92T + 1.55e185T^{2} \) |
| 97 | \( 1 + 3.21e94T + 5.53e188T^{2} \) |
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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
−14.19561358895619809227490086126, −13.62479475345503464121162841721, −11.94768772225357388153019877357, −10.70053352564546293722539708464, −8.114918293821183160691464261656, −6.32856349013046593432052502070, −5.26446172803404771794078880033, −4.07591210979967327433256067865, −2.47078888065588674882906026744, −1.83601962144676027419697295306,
1.83601962144676027419697295306, 2.47078888065588674882906026744, 4.07591210979967327433256067865, 5.26446172803404771794078880033, 6.32856349013046593432052502070, 8.114918293821183160691464261656, 10.70053352564546293722539708464, 11.94768772225357388153019877357, 13.62479475345503464121162841721, 14.19561358895619809227490086126
