L(s) = 1 | + 2.37·2-s + 3.62·4-s − 2.09·5-s − 1.30·7-s + 3.86·8-s − 4.97·10-s + 4.73·11-s − 2.36·13-s − 3.10·14-s + 1.90·16-s − 4.59·17-s − 1.09·19-s − 7.60·20-s + 11.2·22-s + 23-s − 0.604·25-s − 5.60·26-s − 4.75·28-s − 29-s − 2.06·31-s − 3.20·32-s − 10.8·34-s + 2.74·35-s − 1.42·37-s − 2.58·38-s − 8.09·40-s − 5.43·41-s + ⋯ |
L(s) = 1 | + 1.67·2-s + 1.81·4-s − 0.937·5-s − 0.495·7-s + 1.36·8-s − 1.57·10-s + 1.42·11-s − 0.655·13-s − 0.830·14-s + 0.476·16-s − 1.11·17-s − 0.250·19-s − 1.70·20-s + 2.39·22-s + 0.208·23-s − 0.120·25-s − 1.09·26-s − 0.898·28-s − 0.185·29-s − 0.370·31-s − 0.566·32-s − 1.86·34-s + 0.464·35-s − 0.234·37-s − 0.420·38-s − 1.28·40-s − 0.848·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 + 2.09T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + 1.42T + 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 - 3.95T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 0.349T + 59T^{2} \) |
| 61 | \( 1 + 3.89T + 61T^{2} \) |
| 67 | \( 1 + 4.91T + 67T^{2} \) |
| 71 | \( 1 - 0.192T + 71T^{2} \) |
| 73 | \( 1 + 0.0751T + 73T^{2} \) |
| 79 | \( 1 + 7.88T + 79T^{2} \) |
| 83 | \( 1 + 3.95T + 83T^{2} \) |
| 89 | \( 1 + 8.32T + 89T^{2} \) |
| 97 | \( 1 + 7.82T + 97T^{2} \) |
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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
−7.21163321216009537201617121242, −6.91948776327273672256445986547, −6.26464838744875047749639971821, −5.46417711694667072607753478829, −4.58251626928841017860847156101, −4.07968342228253835094787563988, −3.55050919453498436248794989770, −2.71246396036865096645151339676, −1.71220774411483327035050694264, 0,
1.71220774411483327035050694264, 2.71246396036865096645151339676, 3.55050919453498436248794989770, 4.07968342228253835094787563988, 4.58251626928841017860847156101, 5.46417711694667072607753478829, 6.26464838744875047749639971821, 6.91948776327273672256445986547, 7.21163321216009537201617121242