| L(s) = 1 | + 2.67·2-s − 3-s + 5.13·4-s − 2.67·6-s − 7-s + 8.38·8-s + 9-s − 11-s − 5.13·12-s − 0.475·13-s − 2.67·14-s + 12.1·16-s − 1.84·17-s + 2.67·18-s + 2.74·19-s + 21-s − 2.67·22-s + 6.81·23-s − 8.38·24-s − 1.27·26-s − 27-s − 5.13·28-s − 2.58·29-s + 2.82·31-s + 15.6·32-s + 33-s − 4.93·34-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.56·4-s − 1.09·6-s − 0.377·7-s + 2.96·8-s + 0.333·9-s − 0.301·11-s − 1.48·12-s − 0.131·13-s − 0.714·14-s + 3.03·16-s − 0.447·17-s + 0.629·18-s + 0.629·19-s + 0.218·21-s − 0.569·22-s + 1.42·23-s − 1.71·24-s − 0.249·26-s − 0.192·27-s − 0.971·28-s − 0.480·29-s + 0.507·31-s + 2.76·32-s + 0.174·33-s − 0.846·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.902938965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.902938965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 13 | \( 1 + 0.475T + 13T^{2} \) |
| 17 | \( 1 + 1.84T + 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 3.39T + 41T^{2} \) |
| 43 | \( 1 - 0.349T + 43T^{2} \) |
| 47 | \( 1 - 8.65T + 47T^{2} \) |
| 53 | \( 1 - 8.92T + 53T^{2} \) |
| 59 | \( 1 + 4.86T + 59T^{2} \) |
| 61 | \( 1 + 2.78T + 61T^{2} \) |
| 67 | \( 1 - 3.49T + 67T^{2} \) |
| 71 | \( 1 - 2.52T + 71T^{2} \) |
| 73 | \( 1 - 2.42T + 73T^{2} \) |
| 79 | \( 1 + 5.37T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 1.85T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.49205024624462763596854823862, −7.23851724824535523889486714816, −6.35594694387842161037503158430, −5.85804561508718326014145700176, −5.16513617202190110478114560405, −4.56822562817859541934852726560, −3.85064186478450945443592608689, −2.96959777134258176761668094750, −2.33345773549761467370594415165, −1.02873659857793602572642447059,
1.02873659857793602572642447059, 2.33345773549761467370594415165, 2.96959777134258176761668094750, 3.85064186478450945443592608689, 4.56822562817859541934852726560, 5.16513617202190110478114560405, 5.85804561508718326014145700176, 6.35594694387842161037503158430, 7.23851724824535523889486714816, 7.49205024624462763596854823862
