| L(s) = 1 | − 2·2-s + 4·3-s + 4·4-s − 8·6-s + 7·7-s − 8·8-s − 11·9-s + 5·11-s + 16·12-s − 82·13-s − 14·14-s + 16·16-s + 12·17-s + 22·18-s − 42·19-s + 28·21-s − 10·22-s − 175·23-s − 32·24-s + 164·26-s − 152·27-s + 28·28-s + 29-s + 226·31-s − 32·32-s + 20·33-s − 24·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.769·3-s + 1/2·4-s − 0.544·6-s + 0.377·7-s − 0.353·8-s − 0.407·9-s + 0.137·11-s + 0.384·12-s − 1.74·13-s − 0.267·14-s + 1/4·16-s + 0.171·17-s + 0.288·18-s − 0.507·19-s + 0.290·21-s − 0.0969·22-s − 1.58·23-s − 0.272·24-s + 1.23·26-s − 1.08·27-s + 0.188·28-s + 0.00640·29-s + 1.30·31-s − 0.176·32-s + 0.105·33-s − 0.121·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 5 T + p^{3} T^{2} \) |
| 13 | \( 1 + 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 12 T + p^{3} T^{2} \) |
| 19 | \( 1 + 42 T + p^{3} T^{2} \) |
| 23 | \( 1 + 175 T + p^{3} T^{2} \) |
| 29 | \( 1 - T + p^{3} T^{2} \) |
| 31 | \( 1 - 226 T + p^{3} T^{2} \) |
| 37 | \( 1 - 19 T + p^{3} T^{2} \) |
| 41 | \( 1 - 16 T + p^{3} T^{2} \) |
| 43 | \( 1 + 281 T + p^{3} T^{2} \) |
| 47 | \( 1 + 334 T + p^{3} T^{2} \) |
| 53 | \( 1 - 398 T + p^{3} T^{2} \) |
| 59 | \( 1 - 106 T + p^{3} T^{2} \) |
| 61 | \( 1 - 48 T + p^{3} T^{2} \) |
| 67 | \( 1 + 483 T + p^{3} T^{2} \) |
| 71 | \( 1 + 15 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1044 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1253 T + p^{3} T^{2} \) |
| 83 | \( 1 + 758 T + p^{3} T^{2} \) |
| 89 | \( 1 - 86 T + p^{3} T^{2} \) |
| 97 | \( 1 + 710 T + p^{3} T^{2} \) |
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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
−10.23213668055985895392200609320, −9.711288327842076257385822613334, −8.594406256149266428096428131414, −8.005824754100411992588407508639, −7.08729524522785452629692817375, −5.80870337487803572929406072956, −4.44523488470996792926056731342, −2.91528227454680696973931172363, −1.95370671935256496317625120141, 0,
1.95370671935256496317625120141, 2.91528227454680696973931172363, 4.44523488470996792926056731342, 5.80870337487803572929406072956, 7.08729524522785452629692817375, 8.005824754100411992588407508639, 8.594406256149266428096428131414, 9.711288327842076257385822613334, 10.23213668055985895392200609320
