L(s) = 1 | + 2-s − 4-s + 3·7-s − 3·8-s + 11-s − 13-s + 3·14-s − 16-s − 5·17-s − 8·19-s + 22-s − 26-s − 3·28-s − 29-s + 3·31-s + 5·32-s − 5·34-s + 8·37-s − 8·38-s + 2·41-s − 8·43-s − 44-s − 11·47-s + 2·49-s + 52-s − 11·53-s − 9·56-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.13·7-s − 1.06·8-s + 0.301·11-s − 0.277·13-s + 0.801·14-s − 1/4·16-s − 1.21·17-s − 1.83·19-s + 0.213·22-s − 0.196·26-s − 0.566·28-s − 0.185·29-s + 0.538·31-s + 0.883·32-s − 0.857·34-s + 1.31·37-s − 1.29·38-s + 0.312·41-s − 1.21·43-s − 0.150·44-s − 1.60·47-s + 2/7·49-s + 0.138·52-s − 1.51·53-s − 1.20·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.410512419407832359389683993258, −7.79187081368554495796338573590, −6.53501867242475238677795759783, −6.14121902845783398437772376548, −4.83962484932818583295005160761, −4.67701708073401832026802502328, −3.86145623336916765335573728122, −2.67556231321068478617459758001, −1.67746128021095456735981709109, 0,
1.67746128021095456735981709109, 2.67556231321068478617459758001, 3.86145623336916765335573728122, 4.67701708073401832026802502328, 4.83962484932818583295005160761, 6.14121902845783398437772376548, 6.53501867242475238677795759783, 7.79187081368554495796338573590, 8.410512419407832359389683993258