| L(s) = 1 | − i·3-s + (1 − i)5-s − 7-s − i·11-s + (−1 + i)13-s + (−1 − i)15-s + i·21-s + (1 − i)23-s − i·25-s − i·27-s + (−1 − i)29-s − 33-s + (−1 + i)35-s + 37-s + (1 + i)39-s + ⋯ |
| L(s) = 1 | − i·3-s + (1 − i)5-s − 7-s − i·11-s + (−1 + i)13-s + (−1 − i)15-s + i·21-s + (1 − i)23-s − i·25-s − i·27-s + (−1 − i)29-s − 33-s + (−1 + i)35-s + 37-s + (1 + i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.177881353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.177881353\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + iT - T^{2} \) |
| 5 | \( 1 + (-1 + i)T - iT^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 + (1 - i)T - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + (1 + i)T + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 41 | \( 1 + iT - T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 - 2iT - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−8.922880485603564792970134179780, −8.188994542937906160475285769996, −7.16949797029514871679391394019, −6.54996800064447569771127628759, −5.94200494687919938743555753875, −5.06987677481249570320039762999, −4.10494113285929801079505401105, −2.75089738665443547758264968281, −1.92922743326551610108263596566, −0.78077145052131253863577370504,
1.93536883827544989141878694947, 3.02394423742127945405583830380, 3.52166644331070202787119388188, 4.85127538154988837202089334076, 5.36385579259932483480011486118, 6.40017030399039791085852546921, 7.04252880988148823826537993481, 7.70235498441598611791804373852, 9.213562927185144590442048276750, 9.584884824197490572710310952119
