| L(s) = 1 | − 3-s − 2·4-s + 0.399·5-s − 3.44·7-s + 9-s − 4.17·11-s + 2·12-s + 6.21·13-s − 0.399·15-s + 4·16-s + 3.27·17-s + 7.38·19-s − 0.798·20-s + 3.44·21-s + 23-s − 4.84·25-s − 27-s + 6.88·28-s + 29-s − 4.23·31-s + 4.17·33-s − 1.37·35-s − 2·36-s − 4.80·37-s − 6.21·39-s + 2.06·41-s − 0.409·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.178·5-s − 1.30·7-s + 0.333·9-s − 1.25·11-s + 0.577·12-s + 1.72·13-s − 0.103·15-s + 16-s + 0.794·17-s + 1.69·19-s − 0.178·20-s + 0.750·21-s + 0.208·23-s − 0.968·25-s − 0.192·27-s + 1.30·28-s + 0.185·29-s − 0.761·31-s + 0.726·33-s − 0.232·35-s − 0.333·36-s − 0.790·37-s − 0.995·39-s + 0.322·41-s − 0.0623·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 0.399T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 4.17T + 11T^{2} \) |
| 13 | \( 1 - 6.21T + 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 19 | \( 1 - 7.38T + 19T^{2} \) |
| 31 | \( 1 + 4.23T + 31T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 - 2.06T + 41T^{2} \) |
| 43 | \( 1 + 0.409T + 43T^{2} \) |
| 47 | \( 1 + 0.399T + 47T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 + 4.35T + 59T^{2} \) |
| 61 | \( 1 + 8.41T + 61T^{2} \) |
| 67 | \( 1 - 6.88T + 67T^{2} \) |
| 71 | \( 1 + 4.26T + 71T^{2} \) |
| 73 | \( 1 + 2.55T + 73T^{2} \) |
| 79 | \( 1 + 6.29T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.910556351409609309899941037126, −7.982587485741994616598483502096, −7.26635593391893313267965671965, −6.00307246900323079047266751850, −5.73257802446091928327297406455, −4.85399870302378968338975120259, −3.60441707461589379932498327312, −3.17703875575517050058121416814, −1.25411310116159363039347574240, 0,
1.25411310116159363039347574240, 3.17703875575517050058121416814, 3.60441707461589379932498327312, 4.85399870302378968338975120259, 5.73257802446091928327297406455, 6.00307246900323079047266751850, 7.26635593391893313267965671965, 7.982587485741994616598483502096, 8.910556351409609309899941037126
