L(s) = 1 | + 0.414·2-s − 1.82·4-s − 5-s − 7-s − 1.58·8-s − 0.414·10-s + 2.41·11-s − 0.414·13-s − 0.414·14-s + 3·16-s + 0.828·17-s + 4.41·19-s + 1.82·20-s + 0.999·22-s − 4.82·23-s + 25-s − 0.171·26-s + 1.82·28-s + 4·29-s + 6·31-s + 4.41·32-s + 0.343·34-s + 35-s + 8.48·37-s + 1.82·38-s + 1.58·40-s + 2.17·41-s + ⋯ |
L(s) = 1 | + 0.292·2-s − 0.914·4-s − 0.447·5-s − 0.377·7-s − 0.560·8-s − 0.130·10-s + 0.727·11-s − 0.114·13-s − 0.110·14-s + 0.750·16-s + 0.200·17-s + 1.01·19-s + 0.408·20-s + 0.213·22-s − 1.00·23-s + 0.200·25-s − 0.0336·26-s + 0.345·28-s + 0.742·29-s + 1.07·31-s + 0.780·32-s + 0.0588·34-s + 0.169·35-s + 1.39·37-s + 0.296·38-s + 0.250·40-s + 0.339·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.270259422\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.270259422\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 0.414T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 - 4.65T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 2.17T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 + 2.65T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−9.756276417316992312531199280960, −9.399618859337813692756970664894, −8.349047094470360653597024532452, −7.67075274733825648820178862853, −6.49157910502769604567010324267, −5.67617466705980641713739976336, −4.57129197376866602325310791351, −3.88628529066027400245505538327, −2.86280651394709953847482887950, −0.878116091487073529779634856287,
0.878116091487073529779634856287, 2.86280651394709953847482887950, 3.88628529066027400245505538327, 4.57129197376866602325310791351, 5.67617466705980641713739976336, 6.49157910502769604567010324267, 7.67075274733825648820178862853, 8.349047094470360653597024532452, 9.399618859337813692756970664894, 9.756276417316992312531199280960