L(s) = 1 | − 2.45·2-s − 2.96·3-s + 4.00·4-s − 1.96·5-s + 7.25·6-s − 3.26·7-s − 4.91·8-s + 5.77·9-s + 4.81·10-s + 3.56·11-s − 11.8·12-s + 1.09·13-s + 7.99·14-s + 5.82·15-s + 4.02·16-s − 5.29·17-s − 14.1·18-s − 3.97·19-s − 7.87·20-s + 9.66·21-s − 8.72·22-s + 0.162·23-s + 14.5·24-s − 1.13·25-s − 2.67·26-s − 8.22·27-s − 13.0·28-s + ⋯ |
L(s) = 1 | − 1.73·2-s − 1.71·3-s + 2.00·4-s − 0.879·5-s + 2.96·6-s − 1.23·7-s − 1.73·8-s + 1.92·9-s + 1.52·10-s + 1.07·11-s − 3.42·12-s + 0.302·13-s + 2.13·14-s + 1.50·15-s + 1.00·16-s − 1.28·17-s − 3.33·18-s − 0.912·19-s − 1.76·20-s + 2.10·21-s − 1.86·22-s + 0.0338·23-s + 2.96·24-s − 0.226·25-s − 0.523·26-s − 1.58·27-s − 2.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 + 1.96T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 + 3.97T + 19T^{2} \) |
| 23 | \( 1 - 0.162T + 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 + 6.26T + 47T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 1.30T + 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 3.16T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−7.29471928580642852249071305787, −6.79238927682729421338764992664, −6.40470891429211842773886165556, −5.94791682931839911597221250842, −4.55511662376554282232196440500, −4.08556087164511024056322979858, −2.89026922186451645391525875661, −1.64351298874490888493177948935, −0.66718968042959246909871930141, 0,
0.66718968042959246909871930141, 1.64351298874490888493177948935, 2.89026922186451645391525875661, 4.08556087164511024056322979858, 4.55511662376554282232196440500, 5.94791682931839911597221250842, 6.40470891429211842773886165556, 6.79238927682729421338764992664, 7.29471928580642852249071305787