| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 4·11-s − 12-s + 6·13-s + 16-s + 2·17-s − 18-s + 4·19-s + 4·22-s − 8·23-s + 24-s − 6·26-s − 27-s − 2·29-s − 32-s + 4·33-s − 2·34-s + 36-s + 10·37-s − 4·38-s − 6·39-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.852·22-s − 1.66·23-s + 0.204·24-s − 1.17·26-s − 0.192·27-s − 0.371·29-s − 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s + 1.64·37-s − 0.648·38-s − 0.960·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.066237663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.066237663\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.82346428638524176597481330832, −7.52281085391909472304932230967, −6.33754015492779157296733757930, −5.96351299875065035953524843895, −5.35010135518236869010586831192, −4.31089079101385664640002966612, −3.49796811497810747887778433172, −2.58069439638632572608266433034, −1.53767756142141922309330884206, −0.62442247358961096511964059650,
0.62442247358961096511964059650, 1.53767756142141922309330884206, 2.58069439638632572608266433034, 3.49796811497810747887778433172, 4.31089079101385664640002966612, 5.35010135518236869010586831192, 5.96351299875065035953524843895, 6.33754015492779157296733757930, 7.52281085391909472304932230967, 7.82346428638524176597481330832
