| L(s) = 1 | + (−2.23 − 0.114i)5-s + 3.07·7-s − 1.12·11-s + 3.92i·13-s − 0.286i·17-s − 5.12i·19-s + (−4.17 − 2.35i)23-s + (4.97 + 0.510i)25-s − 3.70i·29-s − 5.14·31-s + (−6.86 − 0.351i)35-s + 1.46·37-s − 4.70i·41-s + 4.70·43-s − 3.27·47-s + ⋯ |
| L(s) = 1 | + (−0.998 − 0.0511i)5-s + 1.16·7-s − 0.338·11-s + 1.08i·13-s − 0.0693i·17-s − 1.17i·19-s + (−0.870 − 0.491i)23-s + (0.994 + 0.102i)25-s − 0.688i·29-s − 0.923·31-s + (−1.16 − 0.0593i)35-s + 0.240·37-s − 0.734i·41-s + 0.716·43-s − 0.477·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0502 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0502 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.158701095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.158701095\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.23 + 0.114i)T \) |
| 23 | \( 1 + (4.17 + 2.35i)T \) |
| good | 7 | \( 1 - 3.07T + 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 - 3.92iT - 13T^{2} \) |
| 17 | \( 1 + 0.286iT - 17T^{2} \) |
| 19 | \( 1 + 5.12iT - 19T^{2} \) |
| 29 | \( 1 + 3.70iT - 29T^{2} \) |
| 31 | \( 1 + 5.14T + 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 + 4.70iT - 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 + 3.27T + 47T^{2} \) |
| 53 | \( 1 - 3.68iT - 53T^{2} \) |
| 59 | \( 1 - 1.30iT - 59T^{2} \) |
| 61 | \( 1 - 5.75iT - 61T^{2} \) |
| 67 | \( 1 - 7.13T + 67T^{2} \) |
| 71 | \( 1 - 1.29iT - 71T^{2} \) |
| 73 | \( 1 + 9.83iT - 73T^{2} \) |
| 79 | \( 1 + 0.954iT - 79T^{2} \) |
| 83 | \( 1 + 7.87iT - 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.208287181975983594162770538709, −7.53651498916552442567881765936, −7.00943500128795882013314131147, −6.04337232554808438529702526022, −5.01244512036364654219682711045, −4.47077528337211837422317721699, −3.84691488914664482276444600295, −2.63696954423544604611262440808, −1.73386625263118279754156467177, −0.37232601033408998788828259922,
1.06787952084125082798293528193, 2.16130255700580677502152812281, 3.35950044980489264459158773756, 3.93760550382529267353917925414, 4.94064377646200330535516050539, 5.43997269669430876158119059579, 6.40200572124521323015293819761, 7.47711994460546214413664923504, 7.955851993370218812147770650651, 8.215116476649986872835847589447
