| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 6·11-s − 4·13-s + 15-s + 4·17-s + 8·19-s + 4·21-s + 25-s + 27-s + 2·29-s + 2·31-s − 6·33-s + 4·35-s + 4·37-s − 4·39-s + 6·41-s + 12·43-s + 45-s + 9·49-s + 4·51-s − 14·53-s − 6·55-s + 8·57-s + 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.258·15-s + 0.970·17-s + 1.83·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.359·31-s − 1.04·33-s + 0.676·35-s + 0.657·37-s − 0.640·39-s + 0.937·41-s + 1.82·43-s + 0.149·45-s + 9/7·49-s + 0.560·51-s − 1.92·53-s − 0.809·55-s + 1.05·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.659903858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.659903858\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.325713830979482352772662083545, −8.063378057113881342444489001521, −7.83361019149836895405544176594, −7.25473830199571361426199824321, −5.67296478917078405209824403177, −5.17966513608010269685511467414, −4.48573293025439264559574028699, −2.97997620104463866390256140368, −2.38864408776623314575626052054, −1.15729587229321409219878111918,
1.15729587229321409219878111918, 2.38864408776623314575626052054, 2.97997620104463866390256140368, 4.48573293025439264559574028699, 5.17966513608010269685511467414, 5.67296478917078405209824403177, 7.25473830199571361426199824321, 7.83361019149836895405544176594, 8.063378057113881342444489001521, 9.325713830979482352772662083545
