| L(s) = 1 | + 8.30·3-s + 19.3·5-s + 22.5·7-s + 41.9·9-s − 42.8·11-s + 61.5·13-s + 161.·15-s + 73.0·17-s − 32.1·19-s + 187.·21-s + 23·23-s + 251.·25-s + 124.·27-s − 82.1·29-s − 294.·31-s − 355.·33-s + 438.·35-s + 97.8·37-s + 511.·39-s − 420.·41-s − 97.8·43-s + 814.·45-s − 78.5·47-s + 167.·49-s + 607.·51-s + 501.·53-s − 830.·55-s + ⋯ |
| L(s) = 1 | + 1.59·3-s + 1.73·5-s + 1.21·7-s + 1.55·9-s − 1.17·11-s + 1.31·13-s + 2.77·15-s + 1.04·17-s − 0.388·19-s + 1.94·21-s + 0.208·23-s + 2.01·25-s + 0.887·27-s − 0.526·29-s − 1.70·31-s − 1.87·33-s + 2.11·35-s + 0.434·37-s + 2.09·39-s − 1.60·41-s − 0.346·43-s + 2.69·45-s − 0.243·47-s + 0.487·49-s + 1.66·51-s + 1.30·53-s − 2.03·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.770366500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.770366500\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 8.30T + 27T^{2} \) |
| 5 | \( 1 - 19.3T + 125T^{2} \) |
| 7 | \( 1 - 22.5T + 343T^{2} \) |
| 11 | \( 1 + 42.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 61.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 32.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 82.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 294.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 97.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 420.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 97.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 78.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 501.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 790.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 193.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 562.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 835.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 548.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 164.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 811.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.50e3T + 9.12e5T^{2} \) |
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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
−9.111099065387645863533441697440, −8.303380567764239831060073393531, −7.917111495592987660536524842317, −6.83142978984082485169483660575, −5.63096937141379552815794724682, −5.14896731288608412666787565558, −3.78554279916467140535528337516, −2.83239260500124020672964473612, −1.92081108205958589741405785566, −1.44019368669449011070696046642,
1.44019368669449011070696046642, 1.92081108205958589741405785566, 2.83239260500124020672964473612, 3.78554279916467140535528337516, 5.14896731288608412666787565558, 5.63096937141379552815794724682, 6.83142978984082485169483660575, 7.917111495592987660536524842317, 8.303380567764239831060073393531, 9.111099065387645863533441697440
