| L(s) = 1 | + (−0.143 + 0.0886i)2-s + (−0.273 − 0.961i)3-s + (−0.878 + 1.76i)4-s + (−1.12 + 0.435i)5-s + (0.124 + 0.113i)6-s + (0.127 − 1.37i)7-s + (−0.0616 − 0.665i)8-s + (−0.850 + 0.526i)9-s + (0.122 − 0.162i)10-s + (−3.17 + 1.96i)11-s + (1.93 + 0.362i)12-s + (−0.577 + 6.22i)13-s + (0.103 + 0.207i)14-s + (0.726 + 0.962i)15-s + (−2.30 − 3.05i)16-s + (−3.67 + 3.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.101 + 0.0626i)2-s + (−0.157 − 0.555i)3-s + (−0.439 + 0.882i)4-s + (−0.502 + 0.194i)5-s + (0.0507 + 0.0462i)6-s + (0.0480 − 0.519i)7-s + (−0.0218 − 0.235i)8-s + (−0.283 + 0.175i)9-s + (0.0386 − 0.0512i)10-s + (−0.958 + 0.593i)11-s + (0.559 + 0.104i)12-s + (−0.160 + 1.72i)13-s + (0.0276 + 0.0555i)14-s + (0.187 + 0.248i)15-s + (−0.577 − 0.764i)16-s + (−0.890 + 0.812i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.156007 + 0.403689i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.156007 + 0.403689i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.273 + 0.961i)T \) |
| 103 | \( 1 + (-9.51 + 3.52i)T \) |
| good | 2 | \( 1 + (0.143 - 0.0886i)T + (0.891 - 1.79i)T^{2} \) |
| 5 | \( 1 + (1.12 - 0.435i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (-0.127 + 1.37i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (3.17 - 1.96i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (0.577 - 6.22i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (3.67 - 3.34i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.647 + 2.27i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (4.49 + 2.78i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (-1.05 + 0.409i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (-3.89 - 5.16i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (0.0873 + 0.0163i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (-0.0303 - 0.0117i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (-6.37 + 1.19i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 5.80T + 47T^{2} \) |
| 53 | \( 1 + (-2.05 + 7.21i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (0.379 + 4.10i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (-2.67 + 2.43i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-0.753 - 8.13i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (-3.96 - 1.53i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (9.04 + 3.50i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (6.86 - 2.65i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (-1.39 + 15.1i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (-6.61 - 13.2i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (2.24 + 2.04i)T + (8.95 + 96.5i)T^{2} \) |
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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
−12.04629347692616336895316970235, −11.35604874771033653108392387223, −10.22325686437974049293614694727, −9.016131411966758302042053446693, −8.117439680874649970633611596819, −7.28256275526394308678999500155, −6.57681604446633812739918724794, −4.74148831976721321804606782431, −3.92104971207168188701235601442, −2.27930107304480662756940110766,
0.31899647801245325735018904367, 2.71628084924813837590279332955, 4.28603548392449778760502014162, 5.40239480152983198103876408604, 5.93242090918442474105749741380, 7.79952385803658835820243934299, 8.521766409035135903292496258881, 9.645326636592225182880772511850, 10.34959452098131197820861075345, 11.15028400540139525586559316877
