| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s + 4·11-s + 6·13-s + 16-s − 6·17-s + 8·19-s − 2·20-s + 4·22-s − 25-s + 6·26-s + 6·29-s + 4·31-s + 32-s − 6·34-s + 37-s + 8·38-s − 2·40-s + 6·41-s − 8·43-s + 4·44-s − 8·47-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 1.45·17-s + 1.83·19-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 1.17·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.164·37-s + 1.29·38-s − 0.316·40-s + 0.937·41-s − 1.21·43-s + 0.603·44-s − 1.16·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.224657303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.224657303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−10.89285735420782500834690215413, −9.633028066442567234094648512935, −8.674805511220596933018409082872, −7.87134380463197584693994089861, −6.73345927651178301509646905181, −6.17635572856702991761729027556, −4.79961428857637427378957471382, −3.94279778871499007659572657862, −3.16142223901297670850098979789, −1.32659579994960613230952815319,
1.32659579994960613230952815319, 3.16142223901297670850098979789, 3.94279778871499007659572657862, 4.79961428857637427378957471382, 6.17635572856702991761729027556, 6.73345927651178301509646905181, 7.87134380463197584693994089861, 8.674805511220596933018409082872, 9.633028066442567234094648512935, 10.89285735420782500834690215413
