| L(s) = 1 | − 2.05·2-s + 2.20·4-s − 1.27·7-s − 0.428·8-s + 3.17·11-s + 3.19·13-s + 2.62·14-s − 3.53·16-s − 1.35·17-s + 1.53·19-s − 6.51·22-s − 2.39·23-s − 6.54·26-s − 2.82·28-s + 6.80·29-s − 7.49·31-s + 8.11·32-s + 2.78·34-s − 5.24·37-s − 3.15·38-s + 6.41·41-s − 11.1·43-s + 7.01·44-s + 4.90·46-s − 12.4·47-s − 5.36·49-s + 7.05·52-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 1.10·4-s − 0.483·7-s − 0.151·8-s + 0.957·11-s + 0.885·13-s + 0.701·14-s − 0.884·16-s − 0.329·17-s + 0.352·19-s − 1.38·22-s − 0.498·23-s − 1.28·26-s − 0.534·28-s + 1.26·29-s − 1.34·31-s + 1.43·32-s + 0.477·34-s − 0.861·37-s − 0.511·38-s + 1.00·41-s − 1.69·43-s + 1.05·44-s + 0.723·46-s − 1.81·47-s − 0.766·49-s + 0.977·52-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 | \( 1 + 2.05T + 2T^{2} \) |
| 7 | \( 1 + 1.27T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 + 1.35T + 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 + 2.39T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + 7.49T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 - 6.41T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 0.474T + 53T^{2} \) |
| 59 | \( 1 - 8.04T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 6.45T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 - 1.77T + 73T^{2} \) |
| 79 | \( 1 + 0.781T + 79T^{2} \) |
| 83 | \( 1 - 1.43T + 83T^{2} \) |
| 89 | \( 1 - 7.89T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.073801383784600985859798055920, −7.06374774301566842091754352711, −6.67780429434202389999204273768, −5.95556446839760474992803400402, −4.86834803328720589306126155888, −3.92642978615465797325726866943, −3.14996783093028427206113760931, −1.90902304916640150468498907887, −1.19711527626120232766474143862, 0,
1.19711527626120232766474143862, 1.90902304916640150468498907887, 3.14996783093028427206113760931, 3.92642978615465797325726866943, 4.86834803328720589306126155888, 5.95556446839760474992803400402, 6.67780429434202389999204273768, 7.06374774301566842091754352711, 8.073801383784600985859798055920
