| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 3·9-s + 4·11-s + 6·13-s − 14-s + 16-s − 2·17-s + 3·18-s − 4·22-s − 6·26-s + 28-s + 6·29-s + 8·31-s − 32-s + 2·34-s − 3·36-s + 10·37-s + 2·41-s − 4·43-s + 4·44-s − 8·47-s + 49-s + 6·52-s + 2·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s + 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.852·22-s − 1.17·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 1/2·36-s + 1.64·37-s + 0.312·41-s − 0.609·43-s + 0.603·44-s − 1.16·47-s + 1/7·49-s + 0.832·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.055696301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.055696301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−11.41432333521757255890378761766, −10.68041607351861279913180531310, −9.460094722609108544443052630434, −8.629572853345107589851782221455, −8.089729535357517933440155073370, −6.56718920518778472538358027386, −6.01430128602422585899125610915, −4.36823223624486251008730105784, −2.98711940319117381089884417433, −1.27008084899723889248060652436,
1.27008084899723889248060652436, 2.98711940319117381089884417433, 4.36823223624486251008730105784, 6.01430128602422585899125610915, 6.56718920518778472538358027386, 8.089729535357517933440155073370, 8.629572853345107589851782221455, 9.460094722609108544443052630434, 10.68041607351861279913180531310, 11.41432333521757255890378761766
