| L(s) = 1 | + (−0.100 − 1.41i)2-s + (−3.67 − 0.800i)3-s + (−1.97 + 0.284i)4-s + (4.93 + 0.832i)5-s + (−0.757 + 5.26i)6-s + (3.39 − 6.22i)7-s + (0.601 + 2.76i)8-s + (4.70 + 2.14i)9-s + (0.676 − 7.03i)10-s + (−8.60 − 9.92i)11-s + (7.50 + 0.537i)12-s + (−5.24 − 9.61i)13-s + (−9.12 − 4.16i)14-s + (−17.4 − 7.00i)15-s + (3.83 − 1.12i)16-s + (14.4 + 19.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 − 0.705i)2-s + (−1.22 − 0.266i)3-s + (−0.494 + 0.0711i)4-s + (0.986 + 0.166i)5-s + (−0.126 + 0.878i)6-s + (0.485 − 0.888i)7-s + (0.0751 + 0.345i)8-s + (0.522 + 0.238i)9-s + (0.0676 − 0.703i)10-s + (−0.782 − 0.902i)11-s + (0.625 + 0.0447i)12-s + (−0.403 − 0.739i)13-s + (−0.651 − 0.297i)14-s + (−1.16 − 0.467i)15-s + (0.239 − 0.0704i)16-s + (0.848 + 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0482326 + 0.610687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0482326 + 0.610687i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.100 + 1.41i)T \) |
| 5 | \( 1 + (-4.93 - 0.832i)T \) |
| 23 | \( 1 + (22.9 + 1.55i)T \) |
| good | 3 | \( 1 + (3.67 + 0.800i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (-3.39 + 6.22i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (8.60 + 9.92i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (5.24 + 9.61i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (-14.4 - 19.2i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (27.0 - 3.88i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (52.2 + 7.51i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (15.0 + 9.68i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (22.9 + 61.4i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (12.3 + 27.1i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (2.85 + 0.620i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (-23.1 + 23.1i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-41.5 - 22.7i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (-18.5 + 63.1i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (-54.4 - 34.9i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (2.95 + 41.2i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (1.13 - 1.30i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (78.1 - 104. i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (9.79 - 33.3i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-86.7 + 32.3i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (15.8 + 24.6i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-3.45 - 1.28i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−11.18033425954266280245028532879, −10.61276219235295041672593601371, −10.14947869964148749714694047628, −8.569926285073939335715316899957, −7.41217972932023040828821234514, −5.87370488335787708028332529075, −5.49019353659523192691166475799, −3.84033611965808506585070664540, −1.96997079458970638839756603890, −0.37466772797260635829407328901,
2.10790891838899298994782851332, 4.74348281087590108999877133882, 5.30011670041660904020095261100, 6.13877291468311114381647715263, 7.24255303698993177493112984620, 8.621189715088799933310821123840, 9.662862915189812434069401526989, 10.39607985931534670284356356124, 11.62845924781381324187092509244, 12.37806812041544901778850814166
