L(s) = 1 | + 3-s + 2.60·7-s + 9-s − 4.60·11-s − 4.60·13-s − 2·17-s − 19-s + 2.60·21-s + 2·23-s + 27-s + 2.60·29-s + 4·31-s − 4.60·33-s − 3.39·37-s − 4.60·39-s + 6.60·41-s − 10.6·43-s − 6·47-s − 0.211·49-s − 2·51-s − 57-s + 5.21·59-s − 7.21·61-s + 2.60·63-s − 4·67-s + 2·69-s − 9.21·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.984·7-s + 0.333·9-s − 1.38·11-s − 1.27·13-s − 0.485·17-s − 0.229·19-s + 0.568·21-s + 0.417·23-s + 0.192·27-s + 0.483·29-s + 0.718·31-s − 0.801·33-s − 0.558·37-s − 0.737·39-s + 1.03·41-s − 1.61·43-s − 0.875·47-s − 0.0301·49-s − 0.280·51-s − 0.132·57-s + 0.678·59-s − 0.923·61-s + 0.328·63-s − 0.488·67-s + 0.240·69-s − 1.09·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 3.39T + 37T^{2} \) |
| 41 | \( 1 - 6.60T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 5.21T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 9.21T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 6.60T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.005496798035787714169369289047, −7.21631094202140810468760249785, −6.50850667230466942289921119201, −5.29336658304710231161040098333, −4.92209435464879378526023176096, −4.22655493173202345481604399752, −2.97138795116333100930575535638, −2.46177021673501135857725024930, −1.53899881658505912435319864963, 0,
1.53899881658505912435319864963, 2.46177021673501135857725024930, 2.97138795116333100930575535638, 4.22655493173202345481604399752, 4.92209435464879378526023176096, 5.29336658304710231161040098333, 6.50850667230466942289921119201, 7.21631094202140810468760249785, 8.005496798035787714169369289047