| L(s) = 1 | + 16·2-s + 245.·3-s + 256·4-s − 2.66e3·5-s + 3.92e3·6-s + 4.09e3·8-s + 4.04e4·9-s − 4.26e4·10-s + 6.32e4·11-s + 6.27e4·12-s + 1.47e3·13-s − 6.52e5·15-s + 6.55e4·16-s + 2.00e5·17-s + 6.46e5·18-s + 4.60e5·19-s − 6.81e5·20-s + 1.01e6·22-s − 3.26e5·23-s + 1.00e6·24-s + 5.13e6·25-s + 2.36e4·26-s + 5.08e6·27-s + 1.62e6·29-s − 1.04e7·30-s + 2.95e6·31-s + 1.04e6·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.74·3-s + 0.5·4-s − 1.90·5-s + 1.23·6-s + 0.353·8-s + 2.05·9-s − 1.34·10-s + 1.30·11-s + 0.873·12-s + 0.0143·13-s − 3.32·15-s + 0.250·16-s + 0.581·17-s + 1.45·18-s + 0.810·19-s − 0.952·20-s + 0.921·22-s − 0.243·23-s + 0.617·24-s + 2.62·25-s + 0.0101·26-s + 1.84·27-s + 0.426·29-s − 2.35·30-s + 0.575·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.191509350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.191509350\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 245.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.66e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 6.32e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.47e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.00e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.60e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.26e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.95e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.93e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.07e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.34e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.71e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.88e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.12e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 4.76e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.00e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.86e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.53e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.45e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.97e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.59e8T + 7.60e17T^{2} \) |
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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
−12.15919820200033578764650975352, −11.45183642921469058881493879379, −9.748524991212118834936410659698, −8.492808026178229862731235478807, −7.78589000774604746094362115889, −6.84954474856680528541603170479, −4.46400217660326999443210425749, −3.69934151805079656393346246716, −2.95173666694056566230911881303, −1.17203388790207479149506172589,
1.17203388790207479149506172589, 2.95173666694056566230911881303, 3.69934151805079656393346246716, 4.46400217660326999443210425749, 6.84954474856680528541603170479, 7.78589000774604746094362115889, 8.492808026178229862731235478807, 9.748524991212118834936410659698, 11.45183642921469058881493879379, 12.15919820200033578764650975352
