| L(s) = 1 | + 2·2-s + 4·4-s + 8·5-s + 8·8-s + 16·10-s − 40·11-s − 4·13-s + 16·16-s − 84·17-s − 148·19-s + 32·20-s − 80·22-s − 84·23-s − 61·25-s − 8·26-s − 58·29-s + 136·31-s + 32·32-s − 168·34-s − 222·37-s − 296·38-s + 64·40-s + 420·41-s − 164·43-s − 160·44-s − 168·46-s + 488·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.715·5-s + 0.353·8-s + 0.505·10-s − 1.09·11-s − 0.0853·13-s + 1/4·16-s − 1.19·17-s − 1.78·19-s + 0.357·20-s − 0.775·22-s − 0.761·23-s − 0.487·25-s − 0.0603·26-s − 0.371·29-s + 0.787·31-s + 0.176·32-s − 0.847·34-s − 0.986·37-s − 1.26·38-s + 0.252·40-s + 1.59·41-s − 0.581·43-s − 0.548·44-s − 0.538·46-s + 1.51·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 148 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 420 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 488 T + p^{3} T^{2} \) |
| 53 | \( 1 + 478 T + p^{3} T^{2} \) |
| 59 | \( 1 - 548 T + p^{3} T^{2} \) |
| 61 | \( 1 + 692 T + p^{3} T^{2} \) |
| 67 | \( 1 + 908 T + p^{3} T^{2} \) |
| 71 | \( 1 - 524 T + p^{3} T^{2} \) |
| 73 | \( 1 + 440 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1216 T + p^{3} T^{2} \) |
| 83 | \( 1 + 684 T + p^{3} T^{2} \) |
| 89 | \( 1 - 604 T + p^{3} T^{2} \) |
| 97 | \( 1 - 832 T + p^{3} T^{2} \) |
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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
−9.383069601919496218141170249447, −8.441459164441262383775416098793, −7.54050892498739707596320842340, −6.43592638020726501779468429194, −5.88472260725288840943895918888, −4.83813957202789529665151573269, −4.03377576290121700681513401397, −2.58490831271019271612313648054, −1.95691097407544954216120964807, 0,
1.95691097407544954216120964807, 2.58490831271019271612313648054, 4.03377576290121700681513401397, 4.83813957202789529665151573269, 5.88472260725288840943895918888, 6.43592638020726501779468429194, 7.54050892498739707596320842340, 8.441459164441262383775416098793, 9.383069601919496218141170249447
