L(s) = 1 | − 2·3-s + 2·5-s + 9-s + 4·11-s − 2·13-s − 4·15-s − 2·17-s − 25-s + 4·27-s − 8·29-s + 8·31-s − 8·33-s − 4·37-s + 4·39-s − 10·41-s + 2·43-s + 2·45-s − 7·49-s + 4·51-s + 8·53-s + 8·55-s − 14·59-s − 4·65-s − 8·67-s + 8·71-s + 6·73-s + 2·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 1.03·15-s − 0.485·17-s − 1/5·25-s + 0.769·27-s − 1.48·29-s + 1.43·31-s − 1.39·33-s − 0.657·37-s + 0.640·39-s − 1.56·41-s + 0.304·43-s + 0.298·45-s − 49-s + 0.560·51-s + 1.09·53-s + 1.07·55-s − 1.82·59-s − 0.496·65-s − 0.977·67-s + 0.949·71-s + 0.702·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.76168914480806714380527893619, −6.79557233402867232668771655223, −6.40958411881773003224144743654, −5.75703791119809489827900778094, −5.08240476521979899671599242595, −4.37358730142579141132129987059, −3.34553166164536808016631673380, −2.16636511748180498886542231883, −1.31578059332570785783927581456, 0,
1.31578059332570785783927581456, 2.16636511748180498886542231883, 3.34553166164536808016631673380, 4.37358730142579141132129987059, 5.08240476521979899671599242595, 5.75703791119809489827900778094, 6.40958411881773003224144743654, 6.79557233402867232668771655223, 7.76168914480806714380527893619