| L(s) = 1 | + 1.53·2-s + 0.347·3-s + 1.34·4-s + 1.53·5-s + 0.532·6-s + 7-s + 0.532·8-s − 0.879·9-s + 2.34·10-s + 11-s + 0.467·12-s − 13-s + 1.53·14-s + 0.532·15-s − 0.532·16-s − 1.87·17-s − 1.34·18-s + 2.06·20-s + 0.347·21-s + 1.53·22-s − 23-s + 0.184·24-s + 1.34·25-s − 1.53·26-s − 0.652·27-s + 1.34·28-s + 0.347·29-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 0.347·3-s + 1.34·4-s + 1.53·5-s + 0.532·6-s + 7-s + 0.532·8-s − 0.879·9-s + 2.34·10-s + 11-s + 0.467·12-s − 13-s + 1.53·14-s + 0.532·15-s − 0.532·16-s − 1.87·17-s − 1.34·18-s + 2.06·20-s + 0.347·21-s + 1.53·22-s − 23-s + 0.184·24-s + 1.34·25-s − 1.53·26-s − 0.652·27-s + 1.34·28-s + 0.347·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.970701227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.970701227\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.53T + T^{2} \) |
| 3 | \( 1 - 0.347T + T^{2} \) |
| 5 | \( 1 - 1.53T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - 0.347T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.87T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.87T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.53T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.347T + T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.960069892956419457093284249886, −8.073550374099202900233281497720, −6.86050452637603901286454166791, −6.34953698832712705674546156232, −5.67593993862809497656307995780, −4.91803072298919354356013789907, −4.41140542420386437908004265698, −3.30674401722641486837972125062, −2.13273765115692338867722965639, −2.07383442989205793459930963050,
2.07383442989205793459930963050, 2.13273765115692338867722965639, 3.30674401722641486837972125062, 4.41140542420386437908004265698, 4.91803072298919354356013789907, 5.67593993862809497656307995780, 6.34953698832712705674546156232, 6.86050452637603901286454166791, 8.073550374099202900233281497720, 8.960069892956419457093284249886
