L(s) = 1 | − 3·3-s − 2·5-s − 3·7-s + 6·9-s − 3·11-s + 5·13-s + 6·15-s − 8·17-s − 4·19-s + 9·21-s + 5·23-s − 25-s − 9·27-s + 2·31-s + 9·33-s + 6·35-s + 4·37-s − 15·39-s − 10·41-s + 43-s − 12·45-s + 3·47-s + 2·49-s + 24·51-s − 12·53-s + 6·55-s + 12·57-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.894·5-s − 1.13·7-s + 2·9-s − 0.904·11-s + 1.38·13-s + 1.54·15-s − 1.94·17-s − 0.917·19-s + 1.96·21-s + 1.04·23-s − 1/5·25-s − 1.73·27-s + 0.359·31-s + 1.56·33-s + 1.01·35-s + 0.657·37-s − 2.40·39-s − 1.56·41-s + 0.152·43-s − 1.78·45-s + 0.437·47-s + 2/7·49-s + 3.36·51-s − 1.64·53-s + 0.809·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−6.86056801424133015678012927175, −6.30648419214759360674079453609, −6.05765324497018397512711719092, −4.94866708471461602724793461904, −4.46434431345024855828052664238, −3.72014654214392512365142564104, −2.78509413701889502155122554479, −1.42493498312979270636249615972, 0, 0,
1.42493498312979270636249615972, 2.78509413701889502155122554479, 3.72014654214392512365142564104, 4.46434431345024855828052664238, 4.94866708471461602724793461904, 6.05765324497018397512711719092, 6.30648419214759360674079453609, 6.86056801424133015678012927175