| L(s) = 1 | − 2.16·2-s + 2.70·4-s − 3.32·7-s − 1.53·8-s − 3.82·11-s + 2.65·13-s + 7.21·14-s − 2.08·16-s + 4.73·17-s − 3.65·19-s + 8.28·22-s + 3.61·23-s − 5.75·26-s − 8.99·28-s + 0.0347·29-s − 10.1·31-s + 7.59·32-s − 10.2·34-s + 10.4·37-s + 7.92·38-s + 1.90·41-s − 2.27·43-s − 10.3·44-s − 7.84·46-s + 1.47·47-s + 4.05·49-s + 7.17·52-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.35·4-s − 1.25·7-s − 0.541·8-s − 1.15·11-s + 0.735·13-s + 1.92·14-s − 0.522·16-s + 1.14·17-s − 0.837·19-s + 1.76·22-s + 0.754·23-s − 1.12·26-s − 1.70·28-s + 0.00645·29-s − 1.81·31-s + 1.34·32-s − 1.76·34-s + 1.72·37-s + 1.28·38-s + 0.298·41-s − 0.346·43-s − 1.55·44-s − 1.15·46-s + 0.215·47-s + 0.578·49-s + 0.995·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 7 | \( 1 + 3.32T + 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 - 3.61T + 23T^{2} \) |
| 29 | \( 1 - 0.0347T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 1.90T + 41T^{2} \) |
| 43 | \( 1 + 2.27T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 + 8.92T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 5.17T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 7.24T + 79T^{2} \) |
| 83 | \( 1 + 5.62T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 8.27T + 97T^{2} \) |
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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
−7.85538711840612680460115864263, −7.35593669530334536527265762520, −6.53230878220258492754055178149, −5.93111877931028945063129133562, −5.03113505162088489932563887852, −3.82037530969791099971596629225, −3.01620938619031414713591702901, −2.16683368164966473193157526347, −0.980644608664066387634875765929, 0,
0.980644608664066387634875765929, 2.16683368164966473193157526347, 3.01620938619031414713591702901, 3.82037530969791099971596629225, 5.03113505162088489932563887852, 5.93111877931028945063129133562, 6.53230878220258492754055178149, 7.35593669530334536527265762520, 7.85538711840612680460115864263
