| L(s) = 1 | + 2.21·3-s − 2.22·7-s + 1.88·9-s + 1.57·11-s − 3.96·13-s − 0.294·17-s + 7.76·19-s − 4.91·21-s − 23-s − 2.46·27-s − 9.29·29-s − 9.18·31-s + 3.47·33-s + 10.5·37-s − 8.75·39-s − 2.34·41-s + 6.67·43-s − 1.38·47-s − 2.04·49-s − 0.651·51-s + 11.0·53-s + 17.1·57-s + 5.09·59-s − 8.91·61-s − 4.19·63-s − 1.12·67-s − 2.21·69-s + ⋯ |
| L(s) = 1 | + 1.27·3-s − 0.840·7-s + 0.628·9-s + 0.474·11-s − 1.09·13-s − 0.0715·17-s + 1.78·19-s − 1.07·21-s − 0.208·23-s − 0.473·27-s − 1.72·29-s − 1.65·31-s + 0.605·33-s + 1.73·37-s − 1.40·39-s − 0.365·41-s + 1.01·43-s − 0.201·47-s − 0.292·49-s − 0.0912·51-s + 1.51·53-s + 2.27·57-s + 0.663·59-s − 1.14·61-s − 0.528·63-s − 0.136·67-s − 0.266·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 7 | \( 1 + 2.22T + 7T^{2} \) |
| 11 | \( 1 - 1.57T + 11T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 17 | \( 1 + 0.294T + 17T^{2} \) |
| 19 | \( 1 - 7.76T + 19T^{2} \) |
| 29 | \( 1 + 9.29T + 29T^{2} \) |
| 31 | \( 1 + 9.18T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 2.34T + 41T^{2} \) |
| 43 | \( 1 - 6.67T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 5.09T + 59T^{2} \) |
| 61 | \( 1 + 8.91T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 + 7.60T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 0.199T + 97T^{2} \) |
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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
−7.38400259520165385871209018083, −7.09382920178407571931651486910, −5.88993930930330003779398884631, −5.46226025168838635602518756558, −4.31622583592589782390627758092, −3.66576945956116051618390855162, −3.02342266808229140971462361233, −2.41127248567571471462319650446, −1.44542508104565772106070278343, 0,
1.44542508104565772106070278343, 2.41127248567571471462319650446, 3.02342266808229140971462361233, 3.66576945956116051618390855162, 4.31622583592589782390627758092, 5.46226025168838635602518756558, 5.88993930930330003779398884631, 7.09382920178407571931651486910, 7.38400259520165385871209018083
