L(s) = 1 | + 2.28·2-s + 3.30·3-s + 3.24·4-s + 7.57·6-s + 2.93·7-s + 2.84·8-s + 7.93·9-s − 0.577·11-s + 10.7·12-s − 0.670·13-s + 6.73·14-s + 0.0353·16-s + 0.351·17-s + 18.1·18-s + 9.72·21-s − 1.32·22-s + 4.12·23-s + 9.42·24-s − 1.53·26-s + 16.3·27-s + 9.53·28-s − 3.00·29-s − 0.297·31-s − 5.61·32-s − 1.91·33-s + 0.805·34-s + 25.7·36-s + ⋯ |
L(s) = 1 | + 1.61·2-s + 1.90·3-s + 1.62·4-s + 3.09·6-s + 1.11·7-s + 1.00·8-s + 2.64·9-s − 0.174·11-s + 3.09·12-s − 0.185·13-s + 1.79·14-s + 0.00883·16-s + 0.0853·17-s + 4.28·18-s + 2.12·21-s − 0.281·22-s + 0.859·23-s + 1.92·24-s − 0.301·26-s + 3.14·27-s + 1.80·28-s − 0.558·29-s − 0.0534·31-s − 0.992·32-s − 0.332·33-s + 0.138·34-s + 4.29·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(13.12681524\) |
\(L(\frac12)\) |
\(\approx\) |
\(13.12681524\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 - 2.93T + 7T^{2} \) |
| 11 | \( 1 + 0.577T + 11T^{2} \) |
| 13 | \( 1 + 0.670T + 13T^{2} \) |
| 17 | \( 1 - 0.351T + 17T^{2} \) |
| 23 | \( 1 - 4.12T + 23T^{2} \) |
| 29 | \( 1 + 3.00T + 29T^{2} \) |
| 31 | \( 1 + 0.297T + 31T^{2} \) |
| 37 | \( 1 + 8.30T + 37T^{2} \) |
| 41 | \( 1 + 2.67T + 41T^{2} \) |
| 43 | \( 1 - 6.59T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 4.10T + 53T^{2} \) |
| 59 | \( 1 + 0.610T + 59T^{2} \) |
| 61 | \( 1 - 1.02T + 61T^{2} \) |
| 67 | \( 1 + 3.33T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 7.13T + 73T^{2} \) |
| 79 | \( 1 + 1.54T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 6.57T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.60077922527302154900932351612, −7.16230597118776714452087223553, −6.39178379021488463200377250257, −5.25321491519263858630753359871, −4.82430721831886469283279416405, −4.13807403508048783733616256852, −3.44580918710041739866460621319, −2.90841428474687272981550596901, −2.09264684037084794873596335628, −1.52043261714483481506179689343,
1.52043261714483481506179689343, 2.09264684037084794873596335628, 2.90841428474687272981550596901, 3.44580918710041739866460621319, 4.13807403508048783733616256852, 4.82430721831886469283279416405, 5.25321491519263858630753359871, 6.39178379021488463200377250257, 7.16230597118776714452087223553, 7.60077922527302154900932351612