| L(s) = 1 | + 195.·3-s − 1.27e3·5-s + 2.27e3·7-s + 1.83e4·9-s + 7.17e3·11-s + 2.85e4·13-s − 2.49e5·15-s − 4.47e5·17-s + 5.28e5·19-s + 4.44e5·21-s − 2.24e6·23-s − 3.22e5·25-s − 2.54e5·27-s + 5.98e6·29-s − 1.69e5·31-s + 1.40e6·33-s − 2.90e6·35-s + 1.26e7·37-s + 5.57e6·39-s − 2.76e7·41-s + 2.27e7·43-s − 2.34e7·45-s − 5.32e7·47-s − 3.51e7·49-s − 8.73e7·51-s + 3.18e7·53-s − 9.16e6·55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s − 0.913·5-s + 0.358·7-s + 0.933·9-s + 0.147·11-s + 0.277·13-s − 1.27·15-s − 1.30·17-s + 0.930·19-s + 0.498·21-s − 1.67·23-s − 0.164·25-s − 0.0921·27-s + 1.57·29-s − 0.0329·31-s + 0.205·33-s − 0.327·35-s + 1.10·37-s + 0.385·39-s − 1.52·41-s + 1.01·43-s − 0.853·45-s − 1.59·47-s − 0.871·49-s − 1.80·51-s + 0.554·53-s − 0.135·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 195.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.27e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.27e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.17e3T + 2.35e9T^{2} \) |
| 17 | \( 1 + 4.47e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.28e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.24e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.69e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.26e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.76e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.27e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.32e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.18e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.14e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.80e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.40e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.25e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.88e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.28e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.92e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.14e9T + 7.60e17T^{2} \) |
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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
−10.06758958209399633437277320002, −9.039245587651231746918105105230, −8.181938174080594591620501460676, −7.67709686310137734702313527364, −6.36462749398838407803608042422, −4.61478274580712744175322330520, −3.75829384685768785142009366180, −2.74414690373965449776635639408, −1.58865825813432691043483875373, 0,
1.58865825813432691043483875373, 2.74414690373965449776635639408, 3.75829384685768785142009366180, 4.61478274580712744175322330520, 6.36462749398838407803608042422, 7.67709686310137734702313527364, 8.181938174080594591620501460676, 9.039245587651231746918105105230, 10.06758958209399633437277320002
