| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 4·11-s + 12-s − 4·13-s + 16-s − 3·17-s − 18-s − 6·19-s + 4·22-s + 7·23-s − 24-s + 4·26-s + 27-s + 4·29-s + 5·31-s − 32-s − 4·33-s + 3·34-s + 36-s + 2·37-s + 6·38-s − 4·39-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 − 1.10·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.37·19-s + 0.852·22-s + 1.45·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.742·29-s + 0.898·31-s − 0.176·32-s − 0.696·33-s + 0.514·34-s + 1/6·36-s + 0.328·37-s + 0.973·38-s − 0.640·39-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
| \(L(1)\) |
\(\approx\) |
\(1.178981935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178981935\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.033607291466471403672174684911, −7.27402925322397001645405796860, −6.81646533761944693484572531381, −5.96896575031530914611185550615, −4.89331089757435688711644751472, −4.51431389056753937864892434389, −3.19414529901630591338869336545, −2.57826053496559915505798129114, −1.95414278444254265927284564427, −0.56165559006252591505604011125,
0.56165559006252591505604011125, 1.95414278444254265927284564427, 2.57826053496559915505798129114, 3.19414529901630591338869336545, 4.51431389056753937864892434389, 4.89331089757435688711644751472, 5.96896575031530914611185550615, 6.81646533761944693484572531381, 7.27402925322397001645405796860, 8.033607291466471403672174684911
