| L(s) = 1 | − 3·3-s + 4·5-s + 6·9-s + 11-s + 3·13-s − 12·15-s − 17-s − 2·19-s + 4·23-s + 11·25-s − 9·27-s + 8·31-s − 3·33-s + 8·37-s − 9·39-s + 10·43-s + 24·45-s − 10·47-s + 3·51-s + 3·53-s + 4·55-s + 6·57-s − 14·59-s − 8·61-s + 12·65-s − 10·67-s − 12·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s + 2·9-s + 0.301·11-s + 0.832·13-s − 3.09·15-s − 0.242·17-s − 0.458·19-s + 0.834·23-s + 11/5·25-s − 1.73·27-s + 1.43·31-s − 0.522·33-s + 1.31·37-s − 1.44·39-s + 1.52·43-s + 3.57·45-s − 1.45·47-s + 0.420·51-s + 0.412·53-s + 0.539·55-s + 0.794·57-s − 1.82·59-s − 1.02·61-s + 1.48·65-s − 1.22·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.663820084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.663820084\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.973936402482286354207399587458, −7.69024035678898130149858331901, −6.60165596308849873907420846454, −6.22078735832051372202585427283, −5.87479884156858091839879278168, −4.93772697182014911636657086789, −4.40820492715886183282953413943, −2.87859951484990752955258598597, −1.67453831099288359567119266994, −0.914365510636910812510763279545,
0.914365510636910812510763279545, 1.67453831099288359567119266994, 2.87859951484990752955258598597, 4.40820492715886183282953413943, 4.93772697182014911636657086789, 5.87479884156858091839879278168, 6.22078735832051372202585427283, 6.60165596308849873907420846454, 7.69024035678898130149858331901, 8.973936402482286354207399587458
