| L(s) = 1 | + 4.03·2-s + 8.24·4-s + 8.08·5-s − 5.95·7-s + 0.978·8-s + 32.5·10-s + 17.2·11-s − 23.9·14-s − 61.9·16-s − 92.9·17-s − 13.3·19-s + 66.6·20-s + 69.4·22-s − 219.·23-s − 59.5·25-s − 49.0·28-s + 199.·29-s − 307.·31-s − 257.·32-s − 374.·34-s − 48.1·35-s + 333.·37-s − 53.9·38-s + 7.91·40-s − 200.·41-s + 116.·43-s + 142.·44-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 1.03·4-s + 0.723·5-s − 0.321·7-s + 0.0432·8-s + 1.03·10-s + 0.472·11-s − 0.457·14-s − 0.968·16-s − 1.32·17-s − 0.161·19-s + 0.745·20-s + 0.673·22-s − 1.99·23-s − 0.476·25-s − 0.331·28-s + 1.27·29-s − 1.78·31-s − 1.42·32-s − 1.88·34-s − 0.232·35-s + 1.48·37-s − 0.230·38-s + 0.0312·40-s − 0.764·41-s + 0.414·43-s + 0.486·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.03T + 8T^{2} \) |
| 5 | \( 1 - 8.08T + 125T^{2} \) |
| 7 | \( 1 + 5.95T + 343T^{2} \) |
| 11 | \( 1 - 17.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 92.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 13.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 219.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 199.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 307.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 116.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 26.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 280.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 207.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 285.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 317.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 63.0T + 3.89e5T^{2} \) |
| 79 | \( 1 + 623.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 659.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 603.T + 9.12e5T^{2} \) |
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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
−8.856859941302682481405534902981, −7.70716218712230623671594088037, −6.50630873900597883083194870182, −6.22252155400098607628906105230, −5.38908196581632579700862975569, −4.37202164614312346446761838947, −3.81373992655965531585869808367, −2.61708276212545934325957598665, −1.85117282380160690095556455073, 0,
1.85117282380160690095556455073, 2.61708276212545934325957598665, 3.81373992655965531585869808367, 4.37202164614312346446761838947, 5.38908196581632579700862975569, 6.22252155400098607628906105230, 6.50630873900597883083194870182, 7.70716218712230623671594088037, 8.856859941302682481405534902981
