| L(s) = 1 | + 0.905·2-s − 3.11·3-s − 1.17·4-s − 0.387·5-s − 2.81·6-s − 1.11·7-s − 2.87·8-s + 6.69·9-s − 0.351·10-s + 3.67·12-s − 0.858·13-s − 1.00·14-s + 1.20·15-s − 0.248·16-s + 2.65·17-s + 6.06·18-s + 1.96·19-s + 0.457·20-s + 3.46·21-s − 1.92·23-s + 8.96·24-s − 4.84·25-s − 0.777·26-s − 11.4·27-s + 1.31·28-s − 8.30·29-s + 1.09·30-s + ⋯ |
| L(s) = 1 | + 0.640·2-s − 1.79·3-s − 0.589·4-s − 0.173·5-s − 1.15·6-s − 0.420·7-s − 1.01·8-s + 2.23·9-s − 0.111·10-s + 1.06·12-s − 0.238·13-s − 0.269·14-s + 0.311·15-s − 0.0622·16-s + 0.642·17-s + 1.42·18-s + 0.451·19-s + 0.102·20-s + 0.756·21-s − 0.401·23-s + 1.83·24-s − 0.969·25-s − 0.152·26-s − 2.21·27-s + 0.248·28-s − 1.54·29-s + 0.199·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4146147054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4146147054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 0.905T + 2T^{2} \) |
| 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 + 0.387T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 13 | \( 1 + 0.858T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 - 1.96T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 - 0.0444T + 37T^{2} \) |
| 41 | \( 1 + 0.175T + 41T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 6.13T + 59T^{2} \) |
| 61 | \( 1 + 4.42T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 6.43T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 2.00T + 79T^{2} \) |
| 83 | \( 1 - 0.0835T + 83T^{2} \) |
| 89 | \( 1 + 2.92T + 89T^{2} \) |
| 97 | \( 1 + 5.57T + 97T^{2} \) |
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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
−7.969045447196211499544831207500, −7.29995282197255142347406942482, −6.44985880564624288275145502781, −5.78764383154460100413230735467, −5.43378088085495143548784488274, −4.67093932518472844605349496717, −3.97450814844318269265624777868, −3.23272205698039360104914268804, −1.63536324576877861787539664477, −0.35246287406072615735991926007,
0.35246287406072615735991926007, 1.63536324576877861787539664477, 3.23272205698039360104914268804, 3.97450814844318269265624777868, 4.67093932518472844605349496717, 5.43378088085495143548784488274, 5.78764383154460100413230735467, 6.44985880564624288275145502781, 7.29995282197255142347406942482, 7.969045447196211499544831207500
