| L(s) = 1 | − 9·3-s + 34.8·5-s + 162.·7-s + 81·9-s − 438.·11-s − 538.·13-s − 313.·15-s − 1.14e3·17-s − 1.84e3·19-s − 1.45e3·21-s + 529·23-s − 1.90e3·25-s − 729·27-s + 466.·29-s − 810.·31-s + 3.94e3·33-s + 5.65e3·35-s + 5.99e3·37-s + 4.84e3·39-s + 231.·41-s + 1.01e4·43-s + 2.82e3·45-s + 1.97e4·47-s + 9.49e3·49-s + 1.03e4·51-s − 2.29e4·53-s − 1.53e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.624·5-s + 1.25·7-s + 0.333·9-s − 1.09·11-s − 0.884·13-s − 0.360·15-s − 0.963·17-s − 1.17·19-s − 0.722·21-s + 0.208·23-s − 0.610·25-s − 0.192·27-s + 0.103·29-s − 0.151·31-s + 0.631·33-s + 0.780·35-s + 0.720·37-s + 0.510·39-s + 0.0215·41-s + 0.840·43-s + 0.208·45-s + 1.30·47-s + 0.565·49-s + 0.556·51-s − 1.12·53-s − 0.682·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.642169237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642169237\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 - 529T \) |
| good | 5 | \( 1 - 34.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 162.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 438.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 538.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.84e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 466.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 810.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 231.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.97e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.29e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.50e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.60e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.23e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.45e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 486.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.22e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.94e4T + 8.58e9T^{2} \) |
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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
−9.170989369151688242214436735693, −8.192027606165184481965711041421, −7.53353526936064311365360661556, −6.53302206232359719519827349050, −5.60625271299745627901909129489, −4.91526572095255711354910657017, −4.22275065448686630974600662565, −2.45635261160176503060899564765, −1.93017260742540296630829467469, −0.54569149226523016521751428087,
0.54569149226523016521751428087, 1.93017260742540296630829467469, 2.45635261160176503060899564765, 4.22275065448686630974600662565, 4.91526572095255711354910657017, 5.60625271299745627901909129489, 6.53302206232359719519827349050, 7.53353526936064311365360661556, 8.192027606165184481965711041421, 9.170989369151688242214436735693
