| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 3·11-s + 13-s − 14-s + 16-s + 6·17-s + 2·19-s + 3·22-s + 3·23-s − 5·25-s + 26-s − 28-s + 3·29-s + 5·31-s + 32-s + 6·34-s − 7·37-s + 2·38-s − 6·41-s − 43-s + 3·44-s + 3·46-s − 6·49-s − 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s + 0.639·22-s + 0.625·23-s − 25-s + 0.196·26-s − 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s + 1.02·34-s − 1.15·37-s + 0.324·38-s − 0.937·41-s − 0.152·43-s + 0.452·44-s + 0.442·46-s − 6/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.438724023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.438724023\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−10.40177030669352365127911103866, −9.756591550306942426380153961599, −8.714102464355361131347704225310, −7.68464846057382852123303689608, −6.75704327285852474476313882161, −5.94796173241577251258054909189, −5.00762274654052120160679380267, −3.82098806276952507517625807862, −3.04361032800015499774600610053, −1.39366852032118049822765695687,
1.39366852032118049822765695687, 3.04361032800015499774600610053, 3.82098806276952507517625807862, 5.00762274654052120160679380267, 5.94796173241577251258054909189, 6.75704327285852474476313882161, 7.68464846057382852123303689608, 8.714102464355361131347704225310, 9.756591550306942426380153961599, 10.40177030669352365127911103866
