| L(s) = 1 | − 0.573·2-s − 0.116·3-s − 0.670·4-s + 0.0667·6-s + 1.78·7-s + 0.958·8-s − 0.986·9-s + 0.0780·12-s − 1.02·14-s + 0.121·16-s + 0.565·18-s − 1.37·19-s − 0.207·21-s + 1.19·23-s − 0.111·24-s + 25-s + 0.231·27-s − 1.19·28-s + 1.53·31-s − 1.02·32-s + 0.661·36-s + 0.347·37-s + 0.787·38-s − 1.87·41-s + 0.119·42-s + 1.94·43-s − 0.685·46-s + ⋯ |
| L(s) = 1 | − 0.573·2-s − 0.116·3-s − 0.670·4-s + 0.0667·6-s + 1.78·7-s + 0.958·8-s − 0.986·9-s + 0.0780·12-s − 1.02·14-s + 0.121·16-s + 0.565·18-s − 1.37·19-s − 0.207·21-s + 1.19·23-s − 0.111·24-s + 25-s + 0.231·27-s − 1.19·28-s + 1.53·31-s − 1.02·32-s + 0.661·36-s + 0.347·37-s + 0.787·38-s − 1.87·41-s + 0.119·42-s + 1.94·43-s − 0.685·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7153487601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7153487601\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 - T \) |
| good | 2 | \( 1 + 0.573T + T^{2} \) |
| 3 | \( 1 + 0.116T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 1.78T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.37T + T^{2} \) |
| 23 | \( 1 - 1.19T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.53T + T^{2} \) |
| 37 | \( 1 - 0.347T + T^{2} \) |
| 41 | \( 1 + 1.87T + T^{2} \) |
| 43 | \( 1 - 1.94T + T^{2} \) |
| 47 | \( 1 - 0.792T + T^{2} \) |
| 53 | \( 1 - 1.94T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.67T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.41136320944647271913946753957, −8.974613252401871311646275443655, −8.634663814964282459784361418325, −8.024855018640455403614542103599, −7.06068381465696587115640108548, −5.73511298272627343926057854074, −4.86897047505248595832134159862, −4.27927588831186845009806735243, −2.60375624198040442611285012617, −1.19276263282528018267333375633,
1.19276263282528018267333375633, 2.60375624198040442611285012617, 4.27927588831186845009806735243, 4.86897047505248595832134159862, 5.73511298272627343926057854074, 7.06068381465696587115640108548, 8.024855018640455403614542103599, 8.634663814964282459784361418325, 8.974613252401871311646275443655, 10.41136320944647271913946753957
