| L(s) = 1 | − 1.10·5-s − 1.10·7-s + 5.34·11-s + 4.98·13-s − 2·17-s − 19-s − 2.59·23-s − 3.78·25-s − 4.43·29-s − 8.58·31-s + 1.21·35-s − 2.18·37-s − 5.15·41-s + 2.59·43-s + 2.74·47-s − 5.78·49-s + 9.43·53-s − 5.88·55-s − 3.88·59-s + 2.30·61-s − 5.49·65-s + 5.87·67-s + 9.70·71-s − 8.71·73-s − 5.88·77-s − 3.07·79-s − 12.2·83-s + ⋯ |
| L(s) = 1 | − 0.492·5-s − 0.416·7-s + 1.61·11-s + 1.38·13-s − 0.485·17-s − 0.229·19-s − 0.541·23-s − 0.757·25-s − 0.822·29-s − 1.54·31-s + 0.205·35-s − 0.359·37-s − 0.805·41-s + 0.396·43-s + 0.400·47-s − 0.826·49-s + 1.29·53-s − 0.793·55-s − 0.506·59-s + 0.295·61-s − 0.681·65-s + 0.718·67-s + 1.15·71-s − 1.02·73-s − 0.671·77-s − 0.346·79-s − 1.34·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 1.10T + 5T^{2} \) |
| 7 | \( 1 + 1.10T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 - 4.98T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 23 | \( 1 + 2.59T + 23T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 + 2.18T + 37T^{2} \) |
| 41 | \( 1 + 5.15T + 41T^{2} \) |
| 43 | \( 1 - 2.59T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 - 9.43T + 53T^{2} \) |
| 59 | \( 1 + 3.88T + 59T^{2} \) |
| 61 | \( 1 - 2.30T + 61T^{2} \) |
| 67 | \( 1 - 5.87T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 + 8.71T + 73T^{2} \) |
| 79 | \( 1 + 3.07T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 6.80T + 97T^{2} \) |
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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
−7.32408206723863354520146923304, −6.81691363730694016383859191492, −6.09053252173192762695761679816, −5.60359127610877046405655400294, −4.37457568604466935629701695680, −3.77427712965792262263329303550, −3.47535378043850417483919674931, −2.07133384483723279781941019725, −1.29491082348884553420687775313, 0,
1.29491082348884553420687775313, 2.07133384483723279781941019725, 3.47535378043850417483919674931, 3.77427712965792262263329303550, 4.37457568604466935629701695680, 5.60359127610877046405655400294, 6.09053252173192762695761679816, 6.81691363730694016383859191492, 7.32408206723863354520146923304
