| L(s) = 1 | − 1.08·2-s − 0.820·4-s + 3.80·5-s + 1.68·7-s + 3.06·8-s − 4.12·10-s − 2.23·11-s + 3.12·13-s − 1.82·14-s − 1.68·16-s + 3.40·17-s − 19-s − 3.12·20-s + 2.42·22-s + 7.78·23-s + 9.46·25-s − 3.38·26-s − 1.38·28-s − 2.85·29-s + 5.60·31-s − 4.29·32-s − 3.70·34-s + 6.40·35-s − 0.745·37-s + 1.08·38-s + 11.6·40-s − 4.76·41-s + ⋯ |
| L(s) = 1 | − 0.767·2-s − 0.410·4-s + 1.70·5-s + 0.636·7-s + 1.08·8-s − 1.30·10-s − 0.674·11-s + 0.865·13-s − 0.488·14-s − 0.421·16-s + 0.827·17-s − 0.229·19-s − 0.698·20-s + 0.517·22-s + 1.62·23-s + 1.89·25-s − 0.664·26-s − 0.261·28-s − 0.529·29-s + 1.00·31-s − 0.759·32-s − 0.634·34-s + 1.08·35-s − 0.122·37-s + 0.176·38-s + 1.84·40-s − 0.744·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.129795562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.129795562\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.08T + 2T^{2} \) |
| 5 | \( 1 - 3.80T + 5T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 23 | \( 1 - 7.78T + 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 + 0.745T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 + 9.00T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 - 5.48T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 1.84T + 73T^{2} \) |
| 79 | \( 1 - 7.63T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 - 9.59T + 89T^{2} \) |
| 97 | \( 1 - 2.44T + 97T^{2} \) |
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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
−8.050777290148134939292343654741, −7.25609865446057060594730813734, −6.46182177490310893039877443832, −5.64079747925084132816592551813, −5.17852781963626296851583420142, −4.53015819393063027786447347664, −3.35302907959988056721854222707, −2.40055597265478095129969004587, −1.52370089951462409633284683009, −0.917240203391206051409902021841,
0.917240203391206051409902021841, 1.52370089951462409633284683009, 2.40055597265478095129969004587, 3.35302907959988056721854222707, 4.53015819393063027786447347664, 5.17852781963626296851583420142, 5.64079747925084132816592551813, 6.46182177490310893039877443832, 7.25609865446057060594730813734, 8.050777290148134939292343654741
