| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2.61·7-s + 8-s + 9-s + 3.61·11-s − 12-s − 6.47·13-s + 2.61·14-s + 16-s + 1.23·17-s + 18-s − 5.70·19-s − 2.61·21-s + 3.61·22-s + 4.47·23-s − 24-s − 6.47·26-s − 27-s + 2.61·28-s + 8.47·29-s + 6.61·31-s + 32-s − 3.61·33-s + 1.23·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.989·7-s + 0.353·8-s + 0.333·9-s + 1.09·11-s − 0.288·12-s − 1.79·13-s + 0.699·14-s + 0.250·16-s + 0.299·17-s + 0.235·18-s − 1.30·19-s − 0.571·21-s + 0.771·22-s + 0.932·23-s − 0.204·24-s − 1.26·26-s − 0.192·27-s + 0.494·28-s + 1.57·29-s + 1.18·31-s + 0.176·32-s − 0.629·33-s + 0.211·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.880697696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.880697696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 + 5.61T + 79T^{2} \) |
| 83 | \( 1 - 2.14T + 83T^{2} \) |
| 89 | \( 1 - 3.52T + 89T^{2} \) |
| 97 | \( 1 + 3.38T + 97T^{2} \) |
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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
−8.327642760179491577917520282356, −7.67769673069703016074258279977, −6.76634275037101808728341854434, −6.37497469465247926479730936729, −5.32807157286597180622122201906, −4.56329514094651504224443914125, −4.38983557747224536785818822509, −2.96403950769815336971655070011, −2.07497141696326693390244298711, −0.950612679324047289898337375747,
0.950612679324047289898337375747, 2.07497141696326693390244298711, 2.96403950769815336971655070011, 4.38983557747224536785818822509, 4.56329514094651504224443914125, 5.32807157286597180622122201906, 6.37497469465247926479730936729, 6.76634275037101808728341854434, 7.67769673069703016074258279977, 8.327642760179491577917520282356
