| L(s) = 1 | − i·5-s − 0.245·7-s + 3.43i·11-s + 2.03i·13-s − 4.63i·17-s + (4.02 + 1.67i)19-s + 7.20i·23-s − 25-s − 6.57·29-s − 3.20i·31-s + 0.245i·35-s + 3.48i·37-s + 9.88·41-s + 3.39·43-s − 0.472i·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 0.0926·7-s + 1.03i·11-s + 0.564i·13-s − 1.12i·17-s + (0.923 + 0.383i)19-s + 1.50i·23-s − 0.200·25-s − 1.22·29-s − 0.575i·31-s + 0.0414i·35-s + 0.573i·37-s + 1.54·41-s + 0.517·43-s − 0.0689i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9925753569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9925753569\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 19 | \( 1 + (-4.02 - 1.67i)T \) |
| good | 7 | \( 1 + 0.245T + 7T^{2} \) |
| 11 | \( 1 - 3.43iT - 11T^{2} \) |
| 13 | \( 1 - 2.03iT - 13T^{2} \) |
| 17 | \( 1 + 4.63iT - 17T^{2} \) |
| 23 | \( 1 - 7.20iT - 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 + 3.20iT - 31T^{2} \) |
| 37 | \( 1 - 3.48iT - 37T^{2} \) |
| 41 | \( 1 - 9.88T + 41T^{2} \) |
| 43 | \( 1 - 3.39T + 43T^{2} \) |
| 47 | \( 1 + 0.472iT - 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 + 4.90T + 61T^{2} \) |
| 67 | \( 1 + 9.09iT - 67T^{2} \) |
| 71 | \( 1 + 1.36T + 71T^{2} \) |
| 73 | \( 1 - 1.45T + 73T^{2} \) |
| 79 | \( 1 - 8.90iT - 79T^{2} \) |
| 83 | \( 1 - 3.65iT - 83T^{2} \) |
| 89 | \( 1 + 3.77T + 89T^{2} \) |
| 97 | \( 1 + 1.57iT - 97T^{2} \) |
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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
−7.905984134926284829156921249958, −7.62032240928263558332795476186, −6.93740296581763442617985042815, −6.03986279684181258590465174277, −5.29720242313096481155218845836, −4.69906059685969253219005632658, −3.91240781293778969692556734910, −3.05927495268239515107453485778, −2.02002244396145697724316837821, −1.21759469523957988173750496108,
0.24935175026968480335136546994, 1.42047879713901320952083886692, 2.61148380519925501906401242753, 3.23885138297919691355996453053, 3.99679553407478351963912533064, 4.89972824149321459533101856361, 5.87626607139993619139756422037, 6.12413368646833743021425370613, 7.06458518323650129302237095039, 7.77051411037469439666369192172
