| L(s) = 1 | + (2.30 − 3.11i)2-s + (0.968 − 0.183i)3-s + (−2.06 − 6.70i)4-s + (−11.1 − 1.12i)5-s + (1.65 − 3.44i)6-s + (−18.5 + 0.795i)7-s + (3.59 + 1.25i)8-s + (−24.2 + 9.50i)9-s + (−29.1 + 32.0i)10-s + (−20.1 + 51.2i)11-s + (−3.23 − 6.11i)12-s + (78.9 + 8.89i)13-s + (−40.0 + 59.5i)14-s + (−10.9 + 0.951i)15-s + (58.5 − 39.9i)16-s + (−44.2 − 1.65i)17-s + ⋯ |
| L(s) = 1 | + (0.813 − 1.10i)2-s + (0.186 − 0.0352i)3-s + (−0.258 − 0.838i)4-s + (−0.994 − 0.100i)5-s + (0.112 − 0.234i)6-s + (−0.999 + 0.0429i)7-s + (0.158 + 0.0555i)8-s + (−0.897 + 0.352i)9-s + (−0.920 + 1.01i)10-s + (−0.551 + 1.40i)11-s + (−0.0777 − 0.147i)12-s + (1.68 + 0.189i)13-s + (−0.765 + 1.13i)14-s + (−0.189 + 0.0163i)15-s + (0.915 − 0.624i)16-s + (−0.630 − 0.0236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.625518 + 0.465910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.625518 + 0.465910i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (11.1 + 1.12i)T \) |
| 7 | \( 1 + (18.5 - 0.795i)T \) |
| good | 2 | \( 1 + (-2.30 + 3.11i)T + (-2.35 - 7.64i)T^{2} \) |
| 3 | \( 1 + (-0.968 + 0.183i)T + (25.1 - 9.86i)T^{2} \) |
| 11 | \( 1 + (20.1 - 51.2i)T + (-975. - 905. i)T^{2} \) |
| 13 | \( 1 + (-78.9 - 8.89i)T + (2.14e3 + 488. i)T^{2} \) |
| 17 | \( 1 + (44.2 + 1.65i)T + (4.89e3 + 367. i)T^{2} \) |
| 19 | \( 1 + (37.4 - 64.9i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (4.30 + 115. i)T + (-1.21e4 + 909. i)T^{2} \) |
| 29 | \( 1 + (270. - 61.7i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (161. - 93.0i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (262. - 138. i)T + (2.85e4 - 4.18e4i)T^{2} \) |
| 41 | \( 1 + (-157. - 328. i)T + (-4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (148. + 425. i)T + (-6.21e4 + 4.95e4i)T^{2} \) |
| 47 | \( 1 + (138. + 101. i)T + (3.06e4 + 9.92e4i)T^{2} \) |
| 53 | \( 1 + (96.6 + 51.0i)T + (8.38e4 + 1.23e5i)T^{2} \) |
| 59 | \( 1 + (-1.12 - 15.0i)T + (-2.03e5 + 3.06e4i)T^{2} \) |
| 61 | \( 1 + (-45.7 + 148. i)T + (-1.87e5 - 1.27e5i)T^{2} \) |
| 67 | \( 1 + (-455. - 122. i)T + (2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (8.20 - 35.9i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-270. + 199. i)T + (1.14e5 - 3.71e5i)T^{2} \) |
| 79 | \( 1 + (884. + 510. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-106. - 948. i)T + (-5.57e5 + 1.27e5i)T^{2} \) |
| 89 | \( 1 + (-226. - 576. i)T + (-5.16e5 + 4.79e5i)T^{2} \) |
| 97 | \( 1 + (-304. - 304. i)T + 9.12e5iT^{2} \) |
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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
−11.94548693320548027304400891639, −11.01687983551309937453438483628, −10.41070753453829846160185939156, −8.995188771107288870381895500103, −8.017480582361586153181283496346, −6.76774887666928737470021367021, −5.30072106223188989535050014328, −4.03931260743253285636791941803, −3.31591967052534415011087214087, −1.98022286436307359933339390687,
0.22228612551984119085431505059, 3.32610224409346087788454687338, 3.84028432230343434197238928348, 5.61316416984758392598464811605, 6.17273063636124607088936205864, 7.28985307395877241763173564511, 8.332889851070484677073153343451, 9.060397840570824483128049955005, 10.92543508138917891612414168701, 11.29220579497253045798468345446
