L(s) = 1 | − 2·5-s − 2·7-s − 11-s − 2·13-s − 4·17-s − 6·19-s − 25-s + 8·29-s − 8·31-s + 4·35-s + 10·37-s − 8·41-s − 2·43-s + 8·47-s − 3·49-s + 2·53-s + 2·55-s − 12·59-s + 10·61-s + 4·65-s + 12·67-s − 8·71-s + 6·73-s + 2·77-s − 2·79-s − 16·83-s + 8·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.755·7-s − 0.301·11-s − 0.554·13-s − 0.970·17-s − 1.37·19-s − 1/5·25-s + 1.48·29-s − 1.43·31-s + 0.676·35-s + 1.64·37-s − 1.24·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.269·55-s − 1.56·59-s + 1.28·61-s + 0.496·65-s + 1.46·67-s − 0.949·71-s + 0.702·73-s + 0.227·77-s − 0.225·79-s − 1.75·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.86740863890261993076037568642, −9.989427953048950626922847507675, −8.932199088171823379479580947914, −8.071554028904627902844738284977, −7.05430755180880074285547369845, −6.20097915233518496322100006686, −4.75118710346056158176317605927, −3.79814948526845934439056144160, −2.46483616139600209242493840583, 0,
2.46483616139600209242493840583, 3.79814948526845934439056144160, 4.75118710346056158176317605927, 6.20097915233518496322100006686, 7.05430755180880074285547369845, 8.071554028904627902844738284977, 8.932199088171823379479580947914, 9.989427953048950626922847507675, 10.86740863890261993076037568642