| L(s) = 1 | + 1.10i·2-s + 0.955·3-s + 0.768·4-s + 3.55i·5-s + 1.06i·6-s + i·7-s + 3.07i·8-s − 2.08·9-s − 3.94·10-s − 4.00i·11-s + 0.734·12-s − 1.10·14-s + 3.40i·15-s − 1.87·16-s − 1.86·17-s − 2.31i·18-s + ⋯ |
| L(s) = 1 | + 0.784i·2-s + 0.551·3-s + 0.384·4-s + 1.59i·5-s + 0.433i·6-s + 0.377i·7-s + 1.08i·8-s − 0.695·9-s − 1.24·10-s − 1.20i·11-s + 0.211·12-s − 0.296·14-s + 0.878i·15-s − 0.468·16-s − 0.452·17-s − 0.545i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.905273223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.905273223\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.10iT - 2T^{2} \) |
| 3 | \( 1 - 0.955T + 3T^{2} \) |
| 5 | \( 1 - 3.55iT - 5T^{2} \) |
| 11 | \( 1 + 4.00iT - 11T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 - 6.34iT - 19T^{2} \) |
| 23 | \( 1 + 4.50T + 23T^{2} \) |
| 29 | \( 1 - 8.63T + 29T^{2} \) |
| 31 | \( 1 - 3.22iT - 31T^{2} \) |
| 37 | \( 1 - 1.83iT - 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 0.832iT - 47T^{2} \) |
| 53 | \( 1 - 6.53T + 53T^{2} \) |
| 59 | \( 1 - 5.20iT - 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 + 3.05iT - 67T^{2} \) |
| 71 | \( 1 + 14.4iT - 71T^{2} \) |
| 73 | \( 1 - 12.9iT - 73T^{2} \) |
| 79 | \( 1 - 0.0931T + 79T^{2} \) |
| 83 | \( 1 + 3.17iT - 83T^{2} \) |
| 89 | \( 1 - 12.2iT - 89T^{2} \) |
| 97 | \( 1 - 13.1iT - 97T^{2} \) |
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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
−10.40312706449913686179261612678, −9.018489997673800250671744709652, −8.264481976035209994033682797644, −7.73061536497115609694757832286, −6.73602480302659790429624084630, −6.08555494598856743539435092019, −5.57839574828111028891684023902, −3.75479062039447397153573305979, −2.90252364161518393740127962679, −2.23005646492098090744765146423,
0.71541142661995572640637564504, 1.96607344931169637092855327162, 2.81445920459475136208928608066, 4.20333610936749912147512823006, 4.69413406626220110267709152902, 5.96872241868712401976717102034, 7.03691156349145587658835275624, 7.927837031178577062079269963200, 8.747473227309692208899165636771, 9.451176128961300090333539701946
