| L(s) = 1 | + 266.·3-s − 2.71e3·5-s − 3.41e3·7-s + 5.14e4·9-s − 2.56e3·11-s + 2.85e4·13-s − 7.24e5·15-s − 8.38e4·17-s + 5.90e5·19-s − 9.09e5·21-s + 1.43e6·23-s + 5.42e6·25-s + 8.46e6·27-s − 3.49e6·29-s + 6.50e6·31-s − 6.82e5·33-s + 9.26e6·35-s − 8.53e5·37-s + 7.61e6·39-s + 4.07e6·41-s + 4.43e7·43-s − 1.39e8·45-s + 1.74e6·47-s − 2.87e7·49-s − 2.23e7·51-s + 9.92e7·53-s + 6.95e6·55-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 1.94·5-s − 0.536·7-s + 2.61·9-s − 0.0527·11-s + 0.277·13-s − 3.69·15-s − 0.243·17-s + 1.03·19-s − 1.02·21-s + 1.07·23-s + 2.77·25-s + 3.06·27-s − 0.917·29-s + 1.26·31-s − 0.100·33-s + 1.04·35-s − 0.0748·37-s + 0.527·39-s + 0.225·41-s + 1.97·43-s − 5.07·45-s + 0.0522·47-s − 0.711·49-s − 0.462·51-s + 1.72·53-s + 0.102·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.013446936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.013446936\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 - 266.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.71e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.41e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.56e3T + 2.35e9T^{2} \) |
| 17 | \( 1 + 8.38e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.90e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.43e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.49e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.50e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.53e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.07e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.43e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.74e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.92e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.05e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.10e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.77e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.90e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.41e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 8.73e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.45e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.02e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.54e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.20178430674528981254739421948, −10.90905643576968992045747987119, −9.458371478038318469881642309597, −8.606022549183385588627923851217, −7.71515212998967010675678811036, −7.06823089953085493428743550760, −4.44675257628618225817203061338, −3.54040754129170529729150074303, −2.81040613437810565323173506599, −0.902225964047441609316986844175,
0.902225964047441609316986844175, 2.81040613437810565323173506599, 3.54040754129170529729150074303, 4.44675257628618225817203061338, 7.06823089953085493428743550760, 7.71515212998967010675678811036, 8.606022549183385588627923851217, 9.458371478038318469881642309597, 10.90905643576968992045747987119, 12.20178430674528981254739421948
