| L(s) = 1 | + 1.46·2-s + 0.133·4-s − 0.593·5-s − 2.72·8-s − 0.866·10-s − 4.46·11-s + 4.51·13-s − 4.24·16-s − 0.273·17-s − 2.86·19-s − 0.0789·20-s − 6.51·22-s − 5.05·23-s − 4.64·25-s + 6.59·26-s − 0.352·29-s − 2.51·31-s − 0.751·32-s − 0.399·34-s − 6.64·37-s − 4.18·38-s + 1.61·40-s − 10.8·41-s + 3.38·43-s − 0.593·44-s − 7.38·46-s + 12.4·47-s + ⋯ |
| L(s) = 1 | + 1.03·2-s + 0.0665·4-s − 0.265·5-s − 0.964·8-s − 0.274·10-s − 1.34·11-s + 1.25·13-s − 1.06·16-s − 0.0662·17-s − 0.657·19-s − 0.0176·20-s − 1.38·22-s − 1.05·23-s − 0.929·25-s + 1.29·26-s − 0.0654·29-s − 0.451·31-s − 0.132·32-s − 0.0684·34-s − 1.09·37-s − 0.679·38-s + 0.255·40-s − 1.70·41-s + 0.515·43-s − 0.0894·44-s − 1.08·46-s + 1.81·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 5 | \( 1 + 0.593T + 5T^{2} \) |
| 11 | \( 1 + 4.46T + 11T^{2} \) |
| 13 | \( 1 - 4.51T + 13T^{2} \) |
| 17 | \( 1 + 0.273T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 + 5.05T + 23T^{2} \) |
| 29 | \( 1 + 0.352T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 + 6.64T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 3.38T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 8.05T + 59T^{2} \) |
| 61 | \( 1 - 2.73T + 61T^{2} \) |
| 67 | \( 1 - 5.86T + 67T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + 5.37T + 89T^{2} \) |
| 97 | \( 1 - 2.26T + 97T^{2} \) |
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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
−9.104747239946605214315868426408, −8.376512131616293856483827490991, −7.63680077326841036334339818063, −6.40843831147288304764896740667, −5.76078670978938142690299666510, −4.96004174620579980760948115093, −3.99933726125456773251623483487, −3.32791466629018149422253309057, −2.09371389126008789417815493650, 0,
2.09371389126008789417815493650, 3.32791466629018149422253309057, 3.99933726125456773251623483487, 4.96004174620579980760948115093, 5.76078670978938142690299666510, 6.40843831147288304764896740667, 7.63680077326841036334339818063, 8.376512131616293856483827490991, 9.104747239946605214315868426408
