| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s + 13-s − 15-s − 2·17-s + 4·19-s − 4·23-s + 25-s + 27-s − 6·29-s + 4·31-s − 4·33-s + 6·37-s + 39-s − 2·41-s − 12·43-s − 45-s − 8·47-s − 7·49-s − 2·51-s + 14·53-s + 4·55-s + 4·57-s − 12·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s + 0.986·37-s + 0.160·39-s − 0.312·41-s − 1.82·43-s − 0.149·45-s − 1.16·47-s − 49-s − 0.280·51-s + 1.92·53-s + 0.539·55-s + 0.529·57-s − 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.186345076911240583077837862377, −7.75628303535865087662743738518, −6.99882938289683624510179359507, −6.04354340171130905022563466865, −5.15534121267257844142795039143, −4.37008046601628717192475277567, −3.42036202622825180287678467882, −2.70652455027389045090070848401, −1.60376320281190482583451496239, 0,
1.60376320281190482583451496239, 2.70652455027389045090070848401, 3.42036202622825180287678467882, 4.37008046601628717192475277567, 5.15534121267257844142795039143, 6.04354340171130905022563466865, 6.99882938289683624510179359507, 7.75628303535865087662743738518, 8.186345076911240583077837862377
