L(s) = 1 | − 3-s − 2·5-s + 9-s + 2·11-s + 2·15-s − 2·17-s − 2·23-s − 25-s − 27-s + 6·29-s − 4·31-s − 2·33-s + 6·37-s + 2·41-s − 2·45-s + 2·51-s − 6·53-s − 4·55-s + 12·59-s − 12·61-s + 12·67-s + 2·69-s + 10·71-s − 12·73-s + 75-s − 12·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.603·11-s + 0.516·15-s − 0.485·17-s − 0.417·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.348·33-s + 0.986·37-s + 0.312·41-s − 0.298·45-s + 0.280·51-s − 0.824·53-s − 0.539·55-s + 1.56·59-s − 1.53·61-s + 1.46·67-s + 0.240·69-s + 1.18·71-s − 1.40·73-s + 0.115·75-s − 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.900200107036443557888632457578, −7.19565425199867604846984703457, −6.49144949422330064889655352757, −5.84176384791580753826316565868, −4.85860065633960213694957415344, −4.20350188039766470578075947561, −3.55650103183881699712452674280, −2.40251776418476467417549003132, −1.17821699530441005692002962999, 0,
1.17821699530441005692002962999, 2.40251776418476467417549003132, 3.55650103183881699712452674280, 4.20350188039766470578075947561, 4.85860065633960213694957415344, 5.84176384791580753826316565868, 6.49144949422330064889655352757, 7.19565425199867604846984703457, 7.900200107036443557888632457578