L(s) = 1 | + 2-s − 3.26·3-s + 4-s + 5-s − 3.26·6-s + 2.49·7-s + 8-s + 7.63·9-s + 10-s + 11-s − 3.26·12-s − 4.88·13-s + 2.49·14-s − 3.26·15-s + 16-s + 6.64·17-s + 7.63·18-s + 7.19·19-s + 20-s − 8.14·21-s + 22-s + 7.83·23-s − 3.26·24-s + 25-s − 4.88·26-s − 15.1·27-s + 2.49·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.88·3-s + 0.5·4-s + 0.447·5-s − 1.33·6-s + 0.944·7-s + 0.353·8-s + 2.54·9-s + 0.316·10-s + 0.301·11-s − 0.941·12-s − 1.35·13-s + 0.667·14-s − 0.842·15-s + 0.250·16-s + 1.61·17-s + 1.80·18-s + 1.65·19-s + 0.223·20-s − 1.77·21-s + 0.213·22-s + 1.63·23-s − 0.665·24-s + 0.200·25-s − 0.957·26-s − 2.91·27-s + 0.472·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.308490709\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.308490709\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 3.26T + 3T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 - 7.83T + 23T^{2} \) |
| 29 | \( 1 + 5.75T + 29T^{2} \) |
| 31 | \( 1 + 3.52T + 31T^{2} \) |
| 37 | \( 1 - 7.73T + 37T^{2} \) |
| 41 | \( 1 - 0.928T + 41T^{2} \) |
| 47 | \( 1 + 4.33T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 1.00T + 59T^{2} \) |
| 61 | \( 1 + 4.66T + 61T^{2} \) |
| 67 | \( 1 - 0.388T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 1.87T + 79T^{2} \) |
| 83 | \( 1 + 6.10T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 7.33T + 97T^{2} \) |
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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
−7.77569118408638084538955087681, −7.35237877934778348784858428454, −6.76387936790858612466660816830, −5.69842810356749383021627953417, −5.33522124090973977046202159646, −5.00764511043418960056412291980, −4.12515624611946843123821428056, −2.97718205923144315860761572839, −1.61951553913237081356310170634, −0.919726262935713194546805769075,
0.919726262935713194546805769075, 1.61951553913237081356310170634, 2.97718205923144315860761572839, 4.12515624611946843123821428056, 5.00764511043418960056412291980, 5.33522124090973977046202159646, 5.69842810356749383021627953417, 6.76387936790858612466660816830, 7.35237877934778348784858428454, 7.77569118408638084538955087681