| L(s) = 1 | + (−1.08 + 1.25i)2-s + (−1.71 + 1.10i)3-s + (−0.105 − 0.732i)4-s + (0.415 + 0.909i)5-s + (0.481 − 3.34i)6-s + (−1.10 + 0.324i)7-s + (−1.75 − 1.12i)8-s + (0.489 − 1.07i)9-s + (−1.58 − 0.466i)10-s + (−3.27 − 3.77i)11-s + (0.990 + 1.14i)12-s + (1.49 + 0.438i)13-s + (0.791 − 1.73i)14-s + (−1.71 − 1.10i)15-s + (4.73 − 1.38i)16-s + (−0.641 + 4.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.766 + 0.884i)2-s + (−0.992 + 0.638i)3-s + (−0.0526 − 0.366i)4-s + (0.185 + 0.406i)5-s + (0.196 − 1.36i)6-s + (−0.417 + 0.122i)7-s + (−0.620 − 0.398i)8-s + (0.163 − 0.357i)9-s + (−0.502 − 0.147i)10-s + (−0.987 − 1.13i)11-s + (0.285 + 0.330i)12-s + (0.414 + 0.121i)13-s + (0.211 − 0.463i)14-s + (−0.444 − 0.285i)15-s + (1.18 − 0.347i)16-s + (−0.155 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0671304 - 0.331588i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0671304 - 0.331588i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (3.20 - 3.57i)T \) |
| good | 2 | \( 1 + (1.08 - 1.25i)T + (-0.284 - 1.97i)T^{2} \) |
| 3 | \( 1 + (1.71 - 1.10i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (1.10 - 0.324i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (3.27 + 3.77i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.49 - 0.438i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.641 - 4.46i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.951 - 6.61i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.752 - 5.23i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.728 - 0.468i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.0251 - 0.0550i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.946 - 2.07i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-6.34 + 4.07i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 0.657T + 47T^{2} \) |
| 53 | \( 1 + (-8.80 + 2.58i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (4.65 + 1.36i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.97 - 2.55i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (3.74 - 4.32i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (9.72 - 11.2i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.00 + 6.98i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-4.20 - 1.23i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-4.09 + 8.97i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-11.2 + 7.23i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.737 - 1.61i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−14.44928214620965857739553688827, −13.11321119778829697595066682883, −11.88427951310963934135392598689, −10.68735551473816102743618210526, −10.06344250200927408155026725977, −8.685255870873143135108581066727, −7.68593256584853036116354081195, −6.11052975458295748190197347262, −5.71331811017817401474000844090, −3.55041890082141021713352328408,
0.51554726341548859322484573900, 2.44207304050202547243163608367, 4.97264141966650700930822896254, 6.22451589458177401295536801282, 7.49923769034275714423291798793, 9.043042403499600471111750996293, 9.952640187411211184551365941710, 10.95404612206889881774672948698, 11.84291085895556131761246057784, 12.63062542817127840335726746426
