| L(s) = 1 | + 2·2-s + 8·3-s + 4·4-s + 16·6-s + 7·7-s + 8·8-s + 37·9-s + 68·11-s + 32·12-s − 34·13-s + 14·14-s + 16·16-s − 74·17-s + 74·18-s − 128·19-s + 56·21-s + 136·22-s + 80·23-s + 64·24-s − 68·26-s + 80·27-s + 28·28-s + 286·29-s − 24·31-s + 32·32-s + 544·33-s − 148·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 1/2·4-s + 1.08·6-s + 0.377·7-s + 0.353·8-s + 1.37·9-s + 1.86·11-s + 0.769·12-s − 0.725·13-s + 0.267·14-s + 1/4·16-s − 1.05·17-s + 0.968·18-s − 1.54·19-s + 0.581·21-s + 1.31·22-s + 0.725·23-s + 0.544·24-s − 0.512·26-s + 0.570·27-s + 0.188·28-s + 1.83·29-s − 0.139·31-s + 0.176·32-s + 2.86·33-s − 0.746·34-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
| \(L(2)\) |
\(\approx\) |
\(5.211480454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.211480454\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 68 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 128 T + p^{3} T^{2} \) |
| 23 | \( 1 - 80 T + p^{3} T^{2} \) |
| 29 | \( 1 - 286 T + p^{3} T^{2} \) |
| 31 | \( 1 + 24 T + p^{3} T^{2} \) |
| 37 | \( 1 + 294 T + p^{3} T^{2} \) |
| 41 | \( 1 - 66 T + p^{3} T^{2} \) |
| 43 | \( 1 - 124 T + p^{3} T^{2} \) |
| 47 | \( 1 + 312 T + p^{3} T^{2} \) |
| 53 | \( 1 - 34 T + p^{3} T^{2} \) |
| 59 | \( 1 - 168 T + p^{3} T^{2} \) |
| 61 | \( 1 - 170 T + p^{3} T^{2} \) |
| 67 | \( 1 + 564 T + p^{3} T^{2} \) |
| 71 | \( 1 - 616 T + p^{3} T^{2} \) |
| 73 | \( 1 + 250 T + p^{3} T^{2} \) |
| 79 | \( 1 + 944 T + p^{3} T^{2} \) |
| 83 | \( 1 + 672 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1430 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1270 T + p^{3} T^{2} \) |
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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
−11.18955576209801152537373407883, −10.02646495774156413541411468570, −8.899232880691086136442173117246, −8.500200986133607446452526270200, −7.13494435573915775696510398238, −6.45702957381836343965251844849, −4.64206236789129304092287574140, −3.94117041102020446291137735244, −2.71686229643575911675709249989, −1.67080333129789026443695544768,
1.67080333129789026443695544768, 2.71686229643575911675709249989, 3.94117041102020446291137735244, 4.64206236789129304092287574140, 6.45702957381836343965251844849, 7.13494435573915775696510398238, 8.500200986133607446452526270200, 8.899232880691086136442173117246, 10.02646495774156413541411468570, 11.18955576209801152537373407883
