| L(s) = 1 | + 6.87·3-s − 19.0·5-s + 34.1·7-s + 20.3·9-s + 35.3·11-s + 16.3·13-s − 131.·15-s + 17·17-s − 91.4·19-s + 234.·21-s − 69.1·23-s + 239.·25-s − 45.8·27-s + 72.8·29-s + 314.·31-s + 242.·33-s − 652.·35-s + 103.·37-s + 112.·39-s − 34.8·41-s + 316.·43-s − 388.·45-s + 190.·47-s + 823.·49-s + 116.·51-s − 476.·53-s − 674.·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s − 1.70·5-s + 1.84·7-s + 0.753·9-s + 0.967·11-s + 0.349·13-s − 2.26·15-s + 0.242·17-s − 1.10·19-s + 2.44·21-s − 0.626·23-s + 1.91·25-s − 0.326·27-s + 0.466·29-s + 1.82·31-s + 1.28·33-s − 3.14·35-s + 0.460·37-s + 0.462·39-s − 0.132·41-s + 1.12·43-s − 1.28·45-s + 0.592·47-s + 2.40·49-s + 0.321·51-s − 1.23·53-s − 1.65·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.382162029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.382162029\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 - 6.87T + 27T^{2} \) |
| 5 | \( 1 + 19.0T + 125T^{2} \) |
| 7 | \( 1 - 34.1T + 343T^{2} \) |
| 11 | \( 1 - 35.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 91.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 69.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 72.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 103.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 34.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 316.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 190.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 476.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 351.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 34.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 114.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 611.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 11.4T + 3.89e5T^{2} \) |
| 79 | \( 1 - 266.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 386.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.65e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.40e3T + 9.12e5T^{2} \) |
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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
−8.988231743631033696197193511471, −8.539819838299435322059554841594, −7.893953856108108376076277268474, −7.57388257457410400356660618983, −6.27993786341171113142819572705, −4.56879358802568548700896039940, −4.26954862166127043251788824547, −3.32535601821448820237104257672, −2.12121058956916924059257613589, −0.956143317580032426041578705423,
0.956143317580032426041578705423, 2.12121058956916924059257613589, 3.32535601821448820237104257672, 4.26954862166127043251788824547, 4.56879358802568548700896039940, 6.27993786341171113142819572705, 7.57388257457410400356660618983, 7.893953856108108376076277268474, 8.539819838299435322059554841594, 8.988231743631033696197193511471
