| L(s) = 1 | − 4.82·2-s − 93.8·3-s − 488.·4-s + 1.25e3·5-s + 453.·6-s − 5.92e3·7-s + 4.82e3·8-s − 1.08e4·9-s − 6.07e3·10-s + 2.00e4·11-s + 4.58e4·12-s + 2.85e4·14-s − 1.18e5·15-s + 2.26e5·16-s − 8.95e4·17-s + 5.24e4·18-s − 8.17e5·19-s − 6.15e5·20-s + 5.55e5·21-s − 9.69e4·22-s − 1.17e6·23-s − 4.53e5·24-s − 3.67e5·25-s + 2.86e6·27-s + 2.89e6·28-s + 4.02e6·29-s + 5.70e5·30-s + ⋯ |
| L(s) = 1 | − 0.213·2-s − 0.669·3-s − 0.954·4-s + 0.900·5-s + 0.142·6-s − 0.932·7-s + 0.416·8-s − 0.552·9-s − 0.192·10-s + 0.413·11-s + 0.638·12-s + 0.198·14-s − 0.602·15-s + 0.865·16-s − 0.260·17-s + 0.117·18-s − 1.43·19-s − 0.860·20-s + 0.623·21-s − 0.0882·22-s − 0.875·23-s − 0.278·24-s − 0.188·25-s + 1.03·27-s + 0.889·28-s + 1.05·29-s + 0.128·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.4397925320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4397925320\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 4.82T + 512T^{2} \) |
| 3 | \( 1 + 93.8T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.25e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.92e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.00e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 8.95e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.17e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.17e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.02e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.85e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.56e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 6.67e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.83e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.23e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.54e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.48e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.65e6T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.26e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.95e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.42e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.11e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.77e8T + 7.60e17T^{2} \) |
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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
−10.86195347185550681617397182337, −10.00729177929559867155114990493, −9.205595023986573037861778480440, −8.308815534500516019730632927542, −6.56546406455952793563852519805, −5.90956130722202856178816220991, −4.80364671587985149658426428106, −3.49193683505588953564207401528, −1.89058677250451935453663616616, −0.34460049234107163266200327915,
0.34460049234107163266200327915, 1.89058677250451935453663616616, 3.49193683505588953564207401528, 4.80364671587985149658426428106, 5.90956130722202856178816220991, 6.56546406455952793563852519805, 8.308815534500516019730632927542, 9.205595023986573037861778480440, 10.00729177929559867155114990493, 10.86195347185550681617397182337
