| L(s) = 1 | + i·2-s − 4-s − 1.57·5-s + (0.858 + 2.50i)7-s − i·8-s − 1.57i·10-s + 0.835i·11-s + 4.55i·13-s + (−2.50 + 0.858i)14-s + 16-s − 0.258·17-s − 2.46i·19-s + 1.57·20-s − 0.835·22-s − 5.07i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.703·5-s + (0.324 + 0.945i)7-s − 0.353i·8-s − 0.497i·10-s + 0.251i·11-s + 1.26i·13-s + (−0.668 + 0.229i)14-s + 0.250·16-s − 0.0627·17-s − 0.565i·19-s + 0.351·20-s − 0.178·22-s − 1.05i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6423394015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6423394015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.858 - 2.50i)T \) |
| good | 5 | \( 1 + 1.57T + 5T^{2} \) |
| 11 | \( 1 - 0.835iT - 11T^{2} \) |
| 13 | \( 1 - 4.55iT - 13T^{2} \) |
| 17 | \( 1 + 0.258T + 17T^{2} \) |
| 19 | \( 1 + 2.46iT - 19T^{2} \) |
| 23 | \( 1 + 5.07iT - 23T^{2} \) |
| 29 | \( 1 + 2.91iT - 29T^{2} \) |
| 31 | \( 1 - 9.98iT - 31T^{2} \) |
| 37 | \( 1 - 0.400T + 37T^{2} \) |
| 41 | \( 1 + 9.82T + 41T^{2} \) |
| 43 | \( 1 + 6.15T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 8.19iT - 53T^{2} \) |
| 59 | \( 1 + 1.83T + 59T^{2} \) |
| 61 | \( 1 + 0.703iT - 61T^{2} \) |
| 67 | \( 1 + 8.46T + 67T^{2} \) |
| 71 | \( 1 + 4.76iT - 71T^{2} \) |
| 73 | \( 1 + 7.72iT - 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 5.42T + 83T^{2} \) |
| 89 | \( 1 - 6.03T + 89T^{2} \) |
| 97 | \( 1 - 1.15iT - 97T^{2} \) |
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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
−10.11363522420394090972078514747, −9.129558375278062581305692345990, −8.603793036638256476282550858390, −7.85442915362126075112440585037, −6.85497797628176540210203076076, −6.28648776949958714418010778678, −5.01248383683058912485623958720, −4.50520516616831048434424978103, −3.26199928489739518995734717934, −1.85790726250589606435780068226,
0.28185007888478026375270807429, 1.64822503603174121463584230570, 3.25700688409140287562547585898, 3.80201066295662155414699673251, 4.85678009554721385363189260884, 5.78513273792979701668323436182, 7.07709961182600092704604115331, 7.979971113166223412158911766712, 8.296631424464523976026704521034, 9.719157811124579726143431465398
