| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 3·9-s − 10-s + 4·11-s − 6·13-s − 14-s + 16-s + 2·17-s + 3·18-s + 20-s − 4·22-s + 25-s + 6·26-s + 28-s − 6·29-s + 8·31-s − 32-s − 2·34-s + 35-s − 3·36-s + 10·37-s − 40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s − 0.316·10-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 + 0.223·20-s − 0.852·22-s + 1/5·25-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 + 0.169·35-s − 1/2·36-s + 1.64·37-s − 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.912174345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.912174345\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 43 | \( 1 \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.67637701694952, −12.84062558093450, −12.43460682265429, −11.82177928535394, −11.65446541322723, −11.09436285436015, −10.57750972524545, −9.851550106603068, −9.637815927153413, −9.208687652709070, −8.636937268924191, −8.170605651648051, −7.538516436856556, −7.259155030238261, −6.519597147048690, −6.035911439378080, −5.628661367446109, −4.931179157156950, −4.428347724927151, −3.700721741790909, −2.933327944423806, −2.458270401744897, −1.952897647614620, −1.098195664954170, −0.5171678178603550,
0.5171678178603550, 1.098195664954170, 1.952897647614620, 2.458270401744897, 2.933327944423806, 3.700721741790909, 4.428347724927151, 4.931179157156950, 5.628661367446109, 6.035911439378080, 6.519597147048690, 7.259155030238261, 7.538516436856556, 8.170605651648051, 8.636937268924191, 9.208687652709070, 9.637815927153413, 9.851550106603068, 10.57750972524545, 11.09436285436015, 11.65446541322723, 11.82177928535394, 12.43460682265429, 12.84062558093450, 13.67637701694952
