| L(s) = 1 | + (−2.59 + 0.762i)2-s + (0.775 + 0.894i)3-s + (4.47 − 2.87i)4-s + (−0.142 − 0.989i)5-s + (−2.69 − 1.73i)6-s + (1.26 − 2.77i)7-s + (−5.88 + 6.78i)8-s + (0.227 − 1.58i)9-s + (1.12 + 2.46i)10-s + (3.26 + 0.959i)11-s + (6.04 + 1.77i)12-s + (1.70 + 3.73i)13-s + (−1.17 + 8.18i)14-s + (0.775 − 0.894i)15-s + (5.67 − 12.4i)16-s + (1.12 + 0.721i)17-s + ⋯ |
| L(s) = 1 | + (−1.83 + 0.538i)2-s + (0.447 + 0.516i)3-s + (2.23 − 1.43i)4-s + (−0.0636 − 0.442i)5-s + (−1.09 − 0.706i)6-s + (0.479 − 1.05i)7-s + (−2.07 + 2.40i)8-s + (0.0758 − 0.527i)9-s + (0.355 + 0.778i)10-s + (0.985 + 0.289i)11-s + (1.74 + 0.512i)12-s + (0.473 + 1.03i)13-s + (−0.314 + 2.18i)14-s + (0.200 − 0.230i)15-s + (1.41 − 3.10i)16-s + (0.272 + 0.175i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.590038 + 0.0882934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.590038 + 0.0882934i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (2.73 + 3.94i)T \) |
| good | 2 | \( 1 + (2.59 - 0.762i)T + (1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (-0.775 - 0.894i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-1.26 + 2.77i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-3.26 - 0.959i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.70 - 3.73i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.12 - 0.721i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (1.93 - 1.24i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.53 - 1.62i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (5.45 - 6.29i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.33 + 9.28i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.239 + 1.66i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.82 - 4.41i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 9.13T + 47T^{2} \) |
| 53 | \( 1 + (-0.643 + 1.40i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.222 - 0.488i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (2.97 - 3.43i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (8.98 - 2.63i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (9.15 - 2.68i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (7.17 - 4.61i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-0.573 - 1.25i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.819 - 5.69i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-4.63 - 5.35i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.09 - 7.63i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−14.33396978666366609569415834842, −12.18070832993471370507300300678, −11.02805319167032953174103420034, −10.12320787322611915600062862955, −9.150290602310648429707340313055, −8.553651692600787042126670865013, −7.29109233225274500851386803875, −6.33883736999661060239671398606, −4.13780726561513377347165175869, −1.41620830996076054365820560308,
1.76244149219001788381709558177, 3.04884940805945804544148127912, 6.14969154596760559946484386056, 7.54013890943453528291171194276, 8.262084558925913633126969669647, 9.096310104845860649962191657708, 10.26658359752006937445131027612, 11.31322827484765486468841844963, 11.98246515664557245713809523808, 13.25418005329806916557174103318
