| L(s) = 1 | + 5·3-s − 12·5-s − 11·7-s − 2·9-s + 54·11-s + 11·13-s − 60·15-s − 93·17-s − 19·19-s − 55·21-s − 183·23-s + 19·25-s − 145·27-s − 249·29-s − 56·31-s + 270·33-s + 132·35-s − 250·37-s + 55·39-s + 240·41-s + 196·43-s + 24·45-s + 168·47-s − 222·49-s − 465·51-s + 435·53-s − 648·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 1.07·5-s − 0.593·7-s − 0.0740·9-s + 1.48·11-s + 0.234·13-s − 1.03·15-s − 1.32·17-s − 0.229·19-s − 0.571·21-s − 1.65·23-s + 0.151·25-s − 1.03·27-s − 1.59·29-s − 0.324·31-s + 1.42·33-s + 0.637·35-s − 1.11·37-s + 0.225·39-s + 0.914·41-s + 0.695·43-s + 0.0795·45-s + 0.521·47-s − 0.647·49-s − 1.27·51-s + 1.12·53-s − 1.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + p T \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 11 T + p^{3} T^{2} \) |
| 11 | \( 1 - 54 T + p^{3} T^{2} \) |
| 13 | \( 1 - 11 T + p^{3} T^{2} \) |
| 17 | \( 1 + 93 T + p^{3} T^{2} \) |
| 23 | \( 1 + 183 T + p^{3} T^{2} \) |
| 29 | \( 1 + 249 T + p^{3} T^{2} \) |
| 31 | \( 1 + 56 T + p^{3} T^{2} \) |
| 37 | \( 1 + 250 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 - 196 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 - 435 T + p^{3} T^{2} \) |
| 59 | \( 1 + 195 T + p^{3} T^{2} \) |
| 61 | \( 1 + 358 T + p^{3} T^{2} \) |
| 67 | \( 1 - 961 T + p^{3} T^{2} \) |
| 71 | \( 1 - 246 T + p^{3} T^{2} \) |
| 73 | \( 1 - 353 T + p^{3} T^{2} \) |
| 79 | \( 1 - 34 T + p^{3} T^{2} \) |
| 83 | \( 1 + 234 T + p^{3} T^{2} \) |
| 89 | \( 1 + 168 T + p^{3} T^{2} \) |
| 97 | \( 1 - 758 T + p^{3} T^{2} \) |
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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
−10.99784685254615045194592628480, −9.527011401123062362182140732785, −8.917015998069864571448940636016, −8.060189851905615674275851735522, −7.06576395129521623299493439589, −6.00908300049777687915685225433, −4.04695854338359794693160229896, −3.67130652853172083899920985194, −2.10080515022378018681068411325, 0,
2.10080515022378018681068411325, 3.67130652853172083899920985194, 4.04695854338359794693160229896, 6.00908300049777687915685225433, 7.06576395129521623299493439589, 8.060189851905615674275851735522, 8.917015998069864571448940636016, 9.527011401123062362182140732785, 10.99784685254615045194592628480
