L(s) = 1 | − 2·2-s + 2·4-s + 2·7-s − 2·11-s − 13-s − 4·14-s − 4·16-s − 2·17-s + 4·22-s + 9·23-s + 2·26-s + 4·28-s − 5·29-s + 2·31-s + 8·32-s + 4·34-s − 8·37-s − 12·41-s + 43-s − 4·44-s − 18·46-s + 8·47-s − 3·49-s − 2·52-s − 11·53-s + 10·58-s − 13·61-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s + 0.755·7-s − 0.603·11-s − 0.277·13-s − 1.06·14-s − 16-s − 0.485·17-s + 0.852·22-s + 1.87·23-s + 0.392·26-s + 0.755·28-s − 0.928·29-s + 0.359·31-s + 1.41·32-s + 0.685·34-s − 1.31·37-s − 1.87·41-s + 0.152·43-s − 0.603·44-s − 2.65·46-s + 1.16·47-s − 3/7·49-s − 0.277·52-s − 1.51·53-s + 1.31·58-s − 1.66·61-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 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.528950908624104837677558725985, −7.72070611477353409994103950518, −7.23778612686470899965206676865, −6.42739878412204514749440358407, −5.12771377561824053934514670745, −4.72257746414996013390202582736, −3.32039163045380863323627428423, −2.19303688852089224841075154578, −1.33017423399357553642958647309, 0,
1.33017423399357553642958647309, 2.19303688852089224841075154578, 3.32039163045380863323627428423, 4.72257746414996013390202582736, 5.12771377561824053934514670745, 6.42739878412204514749440358407, 7.23778612686470899965206676865, 7.72070611477353409994103950518, 8.528950908624104837677558725985