| L(s) = 1 | − i·2-s − 4-s + i·7-s + i·8-s − 2·11-s + i·13-s + 14-s + 16-s − 3i·17-s + 2i·22-s − i·23-s + 26-s − i·28-s − 5·29-s + 7·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.377i·7-s + 0.353i·8-s − 0.603·11-s + 0.277i·13-s + 0.267·14-s + 0.250·16-s − 0.727i·17-s + 0.426i·22-s − 0.208i·23-s + 0.196·26-s − 0.188i·28-s − 0.928·29-s + 1.25·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2396439616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2396439616\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 + 5T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 11iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.361993452248401674332663346890, −7.66073612755731953698203906627, −6.75169138636964246514282919024, −5.84186049145859006025153687105, −5.03052056665136633938027836221, −4.35545661260127612915762174986, −3.24293870197874569928238469942, −2.55795096240255478026131980757, −1.52567084440777478986302697144, −0.07440349151446252311737012710,
1.40496130894693070090633102398, 2.76336268455975342400461868637, 3.76247462262771287891751462099, 4.56536236662708124509968998309, 5.43108908830618350397645807681, 6.07461372182509048143826465055, 6.94207816007830559333098286809, 7.59487215809284527637507476696, 8.271283347321822959936361704402, 8.883989964723449463694998060460
