L(s) = 1 | + 2.69·2-s − 3-s + 5.27·4-s − 2.69·6-s + 3.56·7-s + 8.84·8-s + 9-s − 0.0695·11-s − 5.27·12-s − 4.85·13-s + 9.62·14-s + 13.2·16-s − 3.03·17-s + 2.69·18-s + 5.05·19-s − 3.56·21-s − 0.187·22-s + 7.43·23-s − 8.84·24-s − 13.1·26-s − 27-s + 18.8·28-s + 1.95·29-s − 5.63·31-s + 18.1·32-s + 0.0695·33-s − 8.18·34-s + ⋯ |
L(s) = 1 | + 1.90·2-s − 0.577·3-s + 2.63·4-s − 1.10·6-s + 1.34·7-s + 3.12·8-s + 0.333·9-s − 0.0209·11-s − 1.52·12-s − 1.34·13-s + 2.57·14-s + 3.32·16-s − 0.735·17-s + 0.635·18-s + 1.15·19-s − 0.778·21-s − 0.0400·22-s + 1.54·23-s − 1.80·24-s − 2.56·26-s − 0.192·27-s + 3.55·28-s + 0.362·29-s − 1.01·31-s + 3.21·32-s + 0.0121·33-s − 1.40·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.627266978\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.627266978\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 0.0695T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 - 1.95T + 29T^{2} \) |
| 31 | \( 1 + 5.63T + 31T^{2} \) |
| 37 | \( 1 + 6.21T + 37T^{2} \) |
| 41 | \( 1 + 5.63T + 41T^{2} \) |
| 43 | \( 1 + 0.244T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 53 | \( 1 + 8.37T + 53T^{2} \) |
| 59 | \( 1 + 1.60T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 - 7.25T + 71T^{2} \) |
| 73 | \( 1 + 3.69T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 - 8.80T + 83T^{2} \) |
| 89 | \( 1 + 3.55T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−9.348731714254963175778186657765, −8.026677696913516587065328193608, −7.19560953880284008282653765235, −6.80293972975355879667336970765, −5.56956053370863526685052426451, −4.98083410453394385839500661443, −4.72921593852825827925941934472, −3.55540253097291343130710326716, −2.49258455892223997887128397071, −1.50686460071371744456422445347,
1.50686460071371744456422445347, 2.49258455892223997887128397071, 3.55540253097291343130710326716, 4.72921593852825827925941934472, 4.98083410453394385839500661443, 5.56956053370863526685052426451, 6.80293972975355879667336970765, 7.19560953880284008282653765235, 8.026677696913516587065328193608, 9.348731714254963175778186657765