| L(s) = 1 | − 2-s + 1.08·3-s + 4-s + 5-s − 1.08·6-s + 4.96·7-s − 8-s − 1.81·9-s − 10-s − 1.00·11-s + 1.08·12-s − 13-s − 4.96·14-s + 1.08·15-s + 16-s + 5.06·17-s + 1.81·18-s − 4.28·19-s + 20-s + 5.40·21-s + 1.00·22-s − 8.20·23-s − 1.08·24-s + 25-s + 26-s − 5.24·27-s + 4.96·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.629·3-s + 0.5·4-s + 0.447·5-s − 0.444·6-s + 1.87·7-s − 0.353·8-s − 0.604·9-s − 0.316·10-s − 0.301·11-s + 0.314·12-s − 0.277·13-s − 1.32·14-s + 0.281·15-s + 0.250·16-s + 1.22·17-s + 0.427·18-s − 0.983·19-s + 0.223·20-s + 1.17·21-s + 0.213·22-s − 1.71·23-s − 0.222·24-s + 0.200·25-s + 0.196·26-s − 1.00·27-s + 0.937·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.227560043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.227560043\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 1.08T + 3T^{2} \) |
| 7 | \( 1 - 4.96T + 7T^{2} \) |
| 11 | \( 1 + 1.00T + 11T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 4.28T + 19T^{2} \) |
| 23 | \( 1 + 8.20T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 37 | \( 1 - 5.01T + 37T^{2} \) |
| 41 | \( 1 - 7.64T + 41T^{2} \) |
| 43 | \( 1 + 5.56T + 43T^{2} \) |
| 47 | \( 1 - 7.05T + 47T^{2} \) |
| 53 | \( 1 - 9.44T + 53T^{2} \) |
| 59 | \( 1 + 6.34T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 8.15T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 0.894T + 73T^{2} \) |
| 79 | \( 1 - 0.562T + 79T^{2} \) |
| 83 | \( 1 - 7.03T + 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 - 6.54T + 97T^{2} \) |
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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
−8.268841200223260682882090734625, −8.033993149370487712469755216455, −7.38654414374253592466930543615, −6.18205391572029602513265188959, −5.56158313336843012803936799682, −4.72569351028793192853011942146, −3.77403652891635967438230781759, −2.45827894044374215814581520253, −2.11103963521513785413100578763, −0.947118898899582703595084304990,
0.947118898899582703595084304990, 2.11103963521513785413100578763, 2.45827894044374215814581520253, 3.77403652891635967438230781759, 4.72569351028793192853011942146, 5.56158313336843012803936799682, 6.18205391572029602513265188959, 7.38654414374253592466930543615, 8.033993149370487712469755216455, 8.268841200223260682882090734625
