| L(s) = 1 | + 3-s − 4.57·7-s + 9-s + 3.11·11-s + 1.15·13-s − 17-s + 1.34·19-s − 4.57·21-s − 0.615·23-s + 27-s + 3.77·29-s + 2.80·31-s + 3.11·33-s − 11.4·37-s + 1.15·39-s − 0.427·41-s + 10.7·43-s − 10.1·47-s + 13.9·49-s − 51-s + 1.42·53-s + 1.34·57-s − 9.19·59-s + 7.57·61-s − 4.57·63-s + 9.19·67-s − 0.615·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.72·7-s + 0.333·9-s + 0.940·11-s + 0.320·13-s − 0.242·17-s + 0.309·19-s − 0.998·21-s − 0.128·23-s + 0.192·27-s + 0.700·29-s + 0.502·31-s + 0.543·33-s − 1.87·37-s + 0.184·39-s − 0.0668·41-s + 1.63·43-s − 1.47·47-s + 1.98·49-s − 0.140·51-s + 0.195·53-s + 0.178·57-s − 1.19·59-s + 0.970·61-s − 0.576·63-s + 1.12·67-s − 0.0741·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.974131717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.974131717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 19 | \( 1 - 1.34T + 19T^{2} \) |
| 23 | \( 1 + 0.615T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 0.427T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 + 9.19T + 59T^{2} \) |
| 61 | \( 1 - 7.57T + 61T^{2} \) |
| 67 | \( 1 - 9.19T + 67T^{2} \) |
| 71 | \( 1 - 2.30T + 71T^{2} \) |
| 73 | \( 1 - 0.417T + 73T^{2} \) |
| 79 | \( 1 - 7.60T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 1.56T + 89T^{2} \) |
| 97 | \( 1 - 5.84T + 97T^{2} \) |
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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
−8.363011039106929360921708886405, −7.43567317679658575903465051819, −6.55799967869670504738956384499, −6.45820352023750044464812002979, −5.37727874977186462291563137451, −4.29690314952078794500925341441, −3.54036640389071590656976336393, −3.06916405006634620629922921856, −2.00665185784694674208054197020, −0.73606792917685910740423461007,
0.73606792917685910740423461007, 2.00665185784694674208054197020, 3.06916405006634620629922921856, 3.54036640389071590656976336393, 4.29690314952078794500925341441, 5.37727874977186462291563137451, 6.45820352023750044464812002979, 6.55799967869670504738956384499, 7.43567317679658575903465051819, 8.363011039106929360921708886405
