| L(s) = 1 | − 2-s − 3.41·3-s + 4-s + 3.41·6-s + 0.459·7-s − 8-s + 8.66·9-s − 2.03·11-s − 3.41·12-s − 0.459·14-s + 16-s − 5.39·17-s − 8.66·18-s + 3.30·19-s − 1.57·21-s + 2.03·22-s + 4.15·23-s + 3.41·24-s − 19.3·27-s + 0.459·28-s + 6.05·29-s − 6.52·31-s − 32-s + 6.93·33-s + 5.39·34-s + 8.66·36-s + 8.07·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.97·3-s + 0.5·4-s + 1.39·6-s + 0.173·7-s − 0.353·8-s + 2.88·9-s − 0.612·11-s − 0.986·12-s − 0.122·14-s + 0.250·16-s − 1.30·17-s − 2.04·18-s + 0.758·19-s − 0.342·21-s + 0.432·22-s + 0.866·23-s + 0.697·24-s − 3.72·27-s + 0.0868·28-s + 1.12·29-s − 1.17·31-s − 0.176·32-s + 1.20·33-s + 0.924·34-s + 1.44·36-s + 1.32·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3.41T + 3T^{2} \) |
| 7 | \( 1 - 0.459T + 7T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 17 | \( 1 + 5.39T + 17T^{2} \) |
| 19 | \( 1 - 3.30T + 19T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 29 | \( 1 - 6.05T + 29T^{2} \) |
| 31 | \( 1 + 6.52T + 31T^{2} \) |
| 37 | \( 1 - 8.07T + 37T^{2} \) |
| 41 | \( 1 - 4.14T + 41T^{2} \) |
| 43 | \( 1 + 1.26T + 43T^{2} \) |
| 47 | \( 1 - 1.29T + 47T^{2} \) |
| 53 | \( 1 + 5.78T + 53T^{2} \) |
| 59 | \( 1 - 6.77T + 59T^{2} \) |
| 61 | \( 1 + 5.98T + 61T^{2} \) |
| 67 | \( 1 + 8.41T + 67T^{2} \) |
| 71 | \( 1 - 1.47T + 71T^{2} \) |
| 73 | \( 1 + 8.05T + 73T^{2} \) |
| 79 | \( 1 + 1.68T + 79T^{2} \) |
| 83 | \( 1 + 9.99T + 83T^{2} \) |
| 89 | \( 1 + 3.93T + 89T^{2} \) |
| 97 | \( 1 - 7.05T + 97T^{2} \) |
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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
−7.31994538223265138045178489178, −6.73233858437305922063722133413, −6.13141116317103838620395176653, −5.47634477005513653040936015717, −4.80057964702324227104530636271, −4.23939441508911075088934664866, −2.92950755669876126213875639904, −1.78894980190784538463257444149, −0.915363979736705688195134989701, 0,
0.915363979736705688195134989701, 1.78894980190784538463257444149, 2.92950755669876126213875639904, 4.23939441508911075088934664866, 4.80057964702324227104530636271, 5.47634477005513653040936015717, 6.13141116317103838620395176653, 6.73233858437305922063722133413, 7.31994538223265138045178489178
