| L(s) = 1 | + 2.95i·5-s + 0.569i·7-s + 1.70·11-s + 3.55·13-s + 4.91i·17-s + i·19-s + 8.27·23-s − 3.75·25-s + 2.39i·29-s − 4.25i·31-s − 1.68·35-s − 5.95·37-s − 3.11i·41-s − 3.81i·43-s − 5.85·47-s + ⋯ |
| L(s) = 1 | + 1.32i·5-s + 0.215i·7-s + 0.515·11-s + 0.985·13-s + 1.19i·17-s + 0.229i·19-s + 1.72·23-s − 0.750·25-s + 0.444i·29-s − 0.764i·31-s − 0.285·35-s − 0.978·37-s − 0.486i·41-s − 0.582i·43-s − 0.854·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.938157808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.938157808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.95iT - 5T^{2} \) |
| 7 | \( 1 - 0.569iT - 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 - 4.91iT - 17T^{2} \) |
| 23 | \( 1 - 8.27T + 23T^{2} \) |
| 29 | \( 1 - 2.39iT - 29T^{2} \) |
| 31 | \( 1 + 4.25iT - 31T^{2} \) |
| 37 | \( 1 + 5.95T + 37T^{2} \) |
| 41 | \( 1 + 3.11iT - 41T^{2} \) |
| 43 | \( 1 + 3.81iT - 43T^{2} \) |
| 47 | \( 1 + 5.85T + 47T^{2} \) |
| 53 | \( 1 - 4.30iT - 53T^{2} \) |
| 59 | \( 1 - 2.13T + 59T^{2} \) |
| 61 | \( 1 - 4.19T + 61T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 2.29iT - 79T^{2} \) |
| 83 | \( 1 - 1.73T + 83T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 + 5.53T + 97T^{2} \) |
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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
−8.889160562267310654366411739635, −8.395966894485457283954512020524, −7.30439604688129184513861354069, −6.76354770888476242147853046677, −6.10073673330651937759430060958, −5.31151716356007396602666925363, −3.98835132897591755185941190504, −3.43866580362293102071140116041, −2.48657995800330043024403663977, −1.32899435173624581500524745064,
0.71569047341421177728733184087, 1.49588632540874858556952743403, 2.96120401024198455503349043052, 3.90104160291432039968486630460, 4.88412185566366625547616606774, 5.23267728786558350821208233522, 6.40047630752995462439244994885, 7.06202256617185742925512996267, 8.015551400166852960512535844564, 8.842174195813492969560789445818
