| L(s) = 1 | − 3-s − 2.12·5-s + 2.08·7-s + 9-s − 3.99·11-s + 3.23·13-s + 2.12·15-s − 1.33·17-s + 2.48·19-s − 2.08·21-s − 1.08·23-s − 0.482·25-s − 27-s + 5.80·29-s − 9.89·31-s + 3.99·33-s − 4.43·35-s + 7.35·37-s − 3.23·39-s − 10.3·41-s + 5.90·43-s − 2.12·45-s + 11.5·47-s − 2.63·49-s + 1.33·51-s + 5.50·53-s + 8.48·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.950·5-s + 0.789·7-s + 0.333·9-s − 1.20·11-s + 0.898·13-s + 0.548·15-s − 0.324·17-s + 0.570·19-s − 0.455·21-s − 0.227·23-s − 0.0965·25-s − 0.192·27-s + 1.07·29-s − 1.77·31-s + 0.694·33-s − 0.750·35-s + 1.20·37-s − 0.518·39-s − 1.60·41-s + 0.901·43-s − 0.316·45-s + 1.68·47-s − 0.376·49-s + 0.187·51-s + 0.756·53-s + 1.14·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2.12T + 5T^{2} \) |
| 7 | \( 1 - 2.08T + 7T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 + 1.08T + 23T^{2} \) |
| 29 | \( 1 - 5.80T + 29T^{2} \) |
| 31 | \( 1 + 9.89T + 31T^{2} \) |
| 37 | \( 1 - 7.35T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 5.90T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 5.50T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 2.75T + 61T^{2} \) |
| 67 | \( 1 - 8.87T + 67T^{2} \) |
| 71 | \( 1 + 0.828T + 71T^{2} \) |
| 73 | \( 1 + 0.0949T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 8.15T + 83T^{2} \) |
| 89 | \( 1 + 5.10T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.88256580182734383479702175356, −7.59468837848858116832580941642, −6.64455240310993320593431829131, −5.70993014762004750419861175477, −5.11648976239769973124740469031, −4.30773714626138643000602842997, −3.58916438459899756328508685010, −2.46072328029049094524317000605, −1.24245531371421576255507177656, 0,
1.24245531371421576255507177656, 2.46072328029049094524317000605, 3.58916438459899756328508685010, 4.30773714626138643000602842997, 5.11648976239769973124740469031, 5.70993014762004750419861175477, 6.64455240310993320593431829131, 7.59468837848858116832580941642, 7.88256580182734383479702175356
