| L(s) = 1 | − 0.644·2-s − 4.18·3-s − 7.58·4-s + 2.69·6-s + 10.0·8-s − 9.47·9-s − 47.7·11-s + 31.7·12-s + 57.2·13-s + 54.1·16-s − 36.9·17-s + 6.10·18-s − 30.7·19-s + 30.7·22-s − 53.1·23-s − 42.0·24-s − 36.8·26-s + 152.·27-s − 195.·29-s − 257.·31-s − 115.·32-s + 199.·33-s + 23.8·34-s + 71.8·36-s − 346.·37-s + 19.8·38-s − 239.·39-s + ⋯ |
| L(s) = 1 | − 0.227·2-s − 0.805·3-s − 0.948·4-s + 0.183·6-s + 0.443·8-s − 0.350·9-s − 1.30·11-s + 0.763·12-s + 1.22·13-s + 0.846·16-s − 0.527·17-s + 0.0799·18-s − 0.371·19-s + 0.298·22-s − 0.481·23-s − 0.357·24-s − 0.278·26-s + 1.08·27-s − 1.25·29-s − 1.49·31-s − 0.637·32-s + 1.05·33-s + 0.120·34-s + 0.332·36-s − 1.53·37-s + 0.0846·38-s − 0.983·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2474936229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2474936229\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.644T + 8T^{2} \) |
| 3 | \( 1 + 4.18T + 27T^{2} \) |
| 11 | \( 1 + 47.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 36.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 30.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 195.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 257.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 267.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 311.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 98.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 82.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 654.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 779.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 829.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 769.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 613.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 457.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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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
−9.216722549708592987911556345355, −8.594749342262003430766937336339, −7.900063914417561093252622484605, −6.82861860365980969462168490769, −5.63598791150768125345417828845, −5.39395979596312843184408794238, −4.25816024034484253917534093929, −3.29474134888882031349554372212, −1.75236749200196484804577830226, −0.27172789237802276214170255713,
0.27172789237802276214170255713, 1.75236749200196484804577830226, 3.29474134888882031349554372212, 4.25816024034484253917534093929, 5.39395979596312843184408794238, 5.63598791150768125345417828845, 6.82861860365980969462168490769, 7.900063914417561093252622484605, 8.594749342262003430766937336339, 9.216722549708592987911556345355
