| L(s) = 1 | − 264.·3-s − 498.·5-s − 8.28e3·7-s + 5.00e4·9-s − 7.76e4·11-s − 2.85e4·13-s + 1.31e5·15-s − 2.73e5·17-s + 4.36e5·19-s + 2.18e6·21-s − 1.02e6·23-s − 1.70e6·25-s − 8.01e6·27-s + 4.84e6·29-s + 9.42e6·31-s + 2.05e7·33-s + 4.12e6·35-s + 1.33e7·37-s + 7.54e6·39-s − 6.21e6·41-s + 1.50e7·43-s − 2.49e7·45-s − 3.65e7·47-s + 2.82e7·49-s + 7.23e7·51-s + 3.86e7·53-s + 3.86e7·55-s + ⋯ |
| L(s) = 1 | − 1.88·3-s − 0.356·5-s − 1.30·7-s + 2.54·9-s − 1.59·11-s − 0.277·13-s + 0.670·15-s − 0.795·17-s + 0.769·19-s + 2.45·21-s − 0.764·23-s − 0.872·25-s − 2.90·27-s + 1.27·29-s + 1.83·31-s + 3.00·33-s + 0.464·35-s + 1.16·37-s + 0.521·39-s − 0.343·41-s + 0.672·43-s − 0.905·45-s − 1.09·47-s + 0.699·49-s + 1.49·51-s + 0.672·53-s + 0.570·55-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 | \( 1 + 264.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 498.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.28e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.76e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 2.73e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.36e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.02e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.84e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.42e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.33e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 6.21e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.50e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.65e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.86e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.83e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.83e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.77e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.08e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 9.45e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.70e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.12e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.18e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.09e9T + 7.60e17T^{2} \) |
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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
−10.13423688610097035760327238312, −9.948059978777332932150833810110, −7.997295017931871535482652161582, −6.88827878702692421268372454480, −6.12351387615028155201947841419, −5.20517904320852800995217211984, −4.19997785431618932916601146637, −2.62441303640563966304326121304, −0.73044132105071408208733846015, 0,
0.73044132105071408208733846015, 2.62441303640563966304326121304, 4.19997785431618932916601146637, 5.20517904320852800995217211984, 6.12351387615028155201947841419, 6.88827878702692421268372454480, 7.997295017931871535482652161582, 9.948059978777332932150833810110, 10.13423688610097035760327238312
