| L(s) = 1 | + (1.91 + 0.563i)2-s + (1.58 − 1.82i)3-s + (3.36 + 2.16i)4-s + (2.65 − 18.4i)5-s + (4.07 − 2.61i)6-s + (8.33 + 18.2i)7-s + (5.23 + 6.04i)8-s + (3.00 + 20.9i)9-s + (15.4 − 33.9i)10-s + (−45.9 + 13.4i)11-s + (9.29 − 2.72i)12-s + (24.8 − 54.4i)13-s + (5.70 + 39.7i)14-s + (−29.5 − 34.1i)15-s + (6.64 + 14.5i)16-s + (−100. + 64.5i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (0.305 − 0.352i)3-s + (0.420 + 0.270i)4-s + (0.237 − 1.65i)5-s + (0.277 − 0.178i)6-s + (0.449 + 0.985i)7-s + (0.231 + 0.267i)8-s + (0.111 + 0.774i)9-s + (0.489 − 1.07i)10-s + (−1.26 + 0.369i)11-s + (0.223 − 0.0656i)12-s + (0.530 − 1.16i)13-s + (0.108 + 0.758i)14-s + (−0.508 − 0.587i)15-s + (0.103 + 0.227i)16-s + (−1.43 + 0.920i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.269i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.07123 - 0.283899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07123 - 0.283899i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.91 - 0.563i)T \) |
| 23 | \( 1 + (82.6 + 73.0i)T \) |
| good | 3 | \( 1 + (-1.58 + 1.82i)T + (-3.84 - 26.7i)T^{2} \) |
| 5 | \( 1 + (-2.65 + 18.4i)T + (-119. - 35.2i)T^{2} \) |
| 7 | \( 1 + (-8.33 - 18.2i)T + (-224. + 259. i)T^{2} \) |
| 11 | \( 1 + (45.9 - 13.4i)T + (1.11e3 - 719. i)T^{2} \) |
| 13 | \( 1 + (-24.8 + 54.4i)T + (-1.43e3 - 1.66e3i)T^{2} \) |
| 17 | \( 1 + (100. - 64.5i)T + (2.04e3 - 4.46e3i)T^{2} \) |
| 19 | \( 1 + (-52.8 - 33.9i)T + (2.84e3 + 6.23e3i)T^{2} \) |
| 29 | \( 1 + (-3.38 + 2.17i)T + (1.01e4 - 2.21e4i)T^{2} \) |
| 31 | \( 1 + (-115. - 133. i)T + (-4.23e3 + 2.94e4i)T^{2} \) |
| 37 | \( 1 + (-11.8 - 82.3i)T + (-4.86e4 + 1.42e4i)T^{2} \) |
| 41 | \( 1 + (-38.2 + 266. i)T + (-6.61e4 - 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-175. + 202. i)T + (-1.13e4 - 7.86e4i)T^{2} \) |
| 47 | \( 1 - 95.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + (80.6 + 176. i)T + (-9.74e4 + 1.12e5i)T^{2} \) |
| 59 | \( 1 + (44.3 - 97.0i)T + (-1.34e5 - 1.55e5i)T^{2} \) |
| 61 | \( 1 + (475. + 548. i)T + (-3.23e4 + 2.24e5i)T^{2} \) |
| 67 | \( 1 + (289. + 84.9i)T + (2.53e5 + 1.62e5i)T^{2} \) |
| 71 | \( 1 + (-684. - 201. i)T + (3.01e5 + 1.93e5i)T^{2} \) |
| 73 | \( 1 + (-252. - 161. i)T + (1.61e5 + 3.53e5i)T^{2} \) |
| 79 | \( 1 + (21.8 - 47.8i)T + (-3.22e5 - 3.72e5i)T^{2} \) |
| 83 | \( 1 + (138. + 962. i)T + (-5.48e5 + 1.61e5i)T^{2} \) |
| 89 | \( 1 + (306. - 353. i)T + (-1.00e5 - 6.97e5i)T^{2} \) |
| 97 | \( 1 + (-45.1 + 313. i)T + (-8.75e5 - 2.57e5i)T^{2} \) |
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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
−15.47855419352725420316641308565, −13.75459585647950005857528639253, −12.90926238705132473179225765536, −12.33126757899988012135081614210, −10.56926702423511818175294228113, −8.596454594910997825653663988748, −8.011793184412378340825490757553, −5.66998952543789030905947141213, −4.79429637630762672289828696471, −2.12019257383886795530433272512,
2.77091468476067055056487833475, 4.22070715838750840896623590973, 6.35734175164810174064277766938, 7.43161503750379672024403748371, 9.613013590830214707895150880003, 10.82386706234667955205483213901, 11.45853392999157092782837750657, 13.63128993386951899391646403532, 13.94361363905000286226512195952, 15.16400129594322614332734824160
