| L(s) = 1 | + 2·2-s − 9.38·3-s + 4·4-s + 9.38·5-s − 18.7·6-s + 8·8-s + 61·9-s + 18.7·10-s + 20·11-s − 37.5·12-s + 65.6·13-s − 88·15-s + 16·16-s + 56.2·17-s + 122·18-s + 9.38·19-s + 37.5·20-s + 40·22-s + 48·23-s − 75.0·24-s − 37·25-s + 131.·26-s − 318.·27-s − 166·29-s − 176·30-s − 206.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.80·3-s + 0.5·4-s + 0.839·5-s − 1.27·6-s + 0.353·8-s + 2.25·9-s + 0.593·10-s + 0.548·11-s − 0.902·12-s + 1.40·13-s − 1.51·15-s + 0.250·16-s + 0.803·17-s + 1.59·18-s + 0.113·19-s + 0.419·20-s + 0.387·22-s + 0.435·23-s − 0.638·24-s − 0.295·25-s + 0.990·26-s − 2.27·27-s − 1.06·29-s − 1.07·30-s − 1.19·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.678121860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678121860\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 9.38T + 27T^{2} \) |
| 5 | \( 1 - 9.38T + 125T^{2} \) |
| 11 | \( 1 - 20T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 56.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.38T + 6.85e3T^{2} \) |
| 23 | \( 1 - 48T + 1.21e4T^{2} \) |
| 29 | \( 1 + 166T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 78T + 5.06e4T^{2} \) |
| 41 | \( 1 - 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 436T + 7.95e4T^{2} \) |
| 47 | \( 1 - 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 62T + 1.48e5T^{2} \) |
| 59 | \( 1 + 666.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 272.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 580T + 3.00e5T^{2} \) |
| 71 | \( 1 + 544T + 3.57e5T^{2} \) |
| 73 | \( 1 + 600.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 680T + 4.93e5T^{2} \) |
| 83 | \( 1 - 196.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 656.T + 9.12e5T^{2} \) |
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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
−13.18031487263087406742530126025, −12.39851490830612553633858556115, −11.29996879555470691184140180085, −10.69905435151612401309135613329, −9.418810470744951450183158449036, −7.22634835454322682372038762076, −5.96000826408606816820597607154, −5.62322633178832773910526720905, −4.03583657467447267655429694013, −1.33862522665358230379204874497,
1.33862522665358230379204874497, 4.03583657467447267655429694013, 5.62322633178832773910526720905, 5.96000826408606816820597607154, 7.22634835454322682372038762076, 9.418810470744951450183158449036, 10.69905435151612401309135613329, 11.29996879555470691184140180085, 12.39851490830612553633858556115, 13.18031487263087406742530126025
