| L(s) = 1 | + 1.94·2-s + 1.49·3-s + 2.77·4-s − 5-s + 2.90·6-s − 1.13·7-s + 3.43·8-s + 1.24·9-s − 1.94·10-s − 0.709·11-s + 4.14·12-s − 13-s − 2.20·14-s − 1.49·15-s + 3.90·16-s − 1.77·17-s + 2.41·18-s − 2.77·20-s − 1.70·21-s − 1.37·22-s − 0.241·23-s + 5.14·24-s + 25-s − 1.94·26-s + 0.360·27-s − 3.14·28-s − 2.90·30-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 1.49·3-s + 2.77·4-s − 5-s + 2.90·6-s − 1.13·7-s + 3.43·8-s + 1.24·9-s − 1.94·10-s − 0.709·11-s + 4.14·12-s − 13-s − 2.20·14-s − 1.49·15-s + 3.90·16-s − 1.77·17-s + 2.41·18-s − 2.77·20-s − 1.70·21-s − 1.37·22-s − 0.241·23-s + 5.14·24-s + 25-s − 1.94·26-s + 0.360·27-s − 3.14·28-s − 2.90·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.328687810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.328687810\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.94T + T^{2} \) |
| 3 | \( 1 - 1.49T + T^{2} \) |
| 7 | \( 1 + 1.13T + T^{2} \) |
| 11 | \( 1 + 0.709T + T^{2} \) |
| 17 | \( 1 + 1.77T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.241T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.94T + T^{2} \) |
| 47 | \( 1 + 0.241T + T^{2} \) |
| 53 | \( 1 - 0.709T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.709T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.13T + T^{2} \) |
| 97 | \( 1 - 1.49T + T^{2} \) |
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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
−9.291043777197427831127638729434, −8.282864626041522093529867781012, −7.51164134932271194799622555431, −6.97940014708257429482182550836, −6.19642277732035120760566880038, −4.93436465775162315540772188844, −4.22694667553507015242393324442, −3.59290507125894400296651150092, −2.67526900113930576761134083982, −2.41031060249119668675565636561,
2.41031060249119668675565636561, 2.67526900113930576761134083982, 3.59290507125894400296651150092, 4.22694667553507015242393324442, 4.93436465775162315540772188844, 6.19642277732035120760566880038, 6.97940014708257429482182550836, 7.51164134932271194799622555431, 8.282864626041522093529867781012, 9.291043777197427831127638729434
