| L(s) = 1 | + (0.238 − 3.32i)2-s + (−3.71 + 0.808i)3-s + (−3.10 − 0.446i)4-s + (0.457 − 11.1i)5-s + (1.80 + 12.5i)6-s + (−13.4 + 7.33i)7-s + (3.44 − 15.8i)8-s + (−11.4 + 5.21i)9-s + (−37.0 − 4.18i)10-s + (−5.70 − 4.94i)11-s + (11.8 − 0.851i)12-s + (−14.0 + 25.7i)13-s + (21.2 + 46.4i)14-s + (7.32 + 41.8i)15-s + (−76.0 − 22.3i)16-s + (−25.9 − 19.3i)17-s + ⋯ |
| L(s) = 1 | + (0.0841 − 1.17i)2-s + (−0.715 + 0.155i)3-s + (−0.388 − 0.0558i)4-s + (0.0409 − 0.999i)5-s + (0.122 + 0.854i)6-s + (−0.725 + 0.395i)7-s + (0.152 − 0.700i)8-s + (−0.422 + 0.192i)9-s + (−1.17 − 0.132i)10-s + (−0.156 − 0.135i)11-s + (0.286 − 0.0204i)12-s + (−0.299 + 0.548i)13-s + (0.404 + 0.886i)14-s + (0.126 + 0.720i)15-s + (−1.18 − 0.348i)16-s + (−0.369 − 0.276i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.195253 + 0.377972i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.195253 + 0.377972i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.457 + 11.1i)T \) |
| 23 | \( 1 + (-6.97 + 110. i)T \) |
| good | 2 | \( 1 + (-0.238 + 3.32i)T + (-7.91 - 1.13i)T^{2} \) |
| 3 | \( 1 + (3.71 - 0.808i)T + (24.5 - 11.2i)T^{2} \) |
| 7 | \( 1 + (13.4 - 7.33i)T + (185. - 288. i)T^{2} \) |
| 11 | \( 1 + (5.70 + 4.94i)T + (189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (14.0 - 25.7i)T + (-1.18e3 - 1.84e3i)T^{2} \) |
| 17 | \( 1 + (25.9 + 19.3i)T + (1.38e3 + 4.71e3i)T^{2} \) |
| 19 | \( 1 + (22.4 - 156. i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (122. - 17.5i)T + (2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-148. + 95.4i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (134. + 50.0i)T + (3.82e4 + 3.31e4i)T^{2} \) |
| 41 | \( 1 + (-86.5 + 189. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (88.8 + 408. i)T + (-7.23e4 + 3.30e4i)T^{2} \) |
| 47 | \( 1 + (384. + 384. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (154. + 283. i)T + (-8.04e4 + 1.25e5i)T^{2} \) |
| 59 | \( 1 + (-48.5 - 165. i)T + (-1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (235. + 366. i)T + (-9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-940. - 67.2i)T + (2.97e5 + 4.28e4i)T^{2} \) |
| 71 | \( 1 + (-621. - 717. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-207. - 277. i)T + (-1.09e5 + 3.73e5i)T^{2} \) |
| 79 | \( 1 + (-205. + 60.3i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-180. + 482. i)T + (-4.32e5 - 3.74e5i)T^{2} \) |
| 89 | \( 1 + (77.8 + 50.0i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-100. - 268. i)T + (-6.89e5 + 5.97e5i)T^{2} \) |
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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
−12.24844575955123850648295202149, −11.58175883970968908643361422389, −10.43603423030718812529214137036, −9.570690043707023186794799856889, −8.376351464780649077305743567083, −6.48823186593463981815545061494, −5.26445191131355039570315179619, −3.86761165014234025115219772514, −2.12852270066757548014829425513, −0.22427661564042825928982730510,
2.93733434529500778099376279023, 5.03943852864589644387919995944, 6.31969511681245534109726591593, 6.77982395492285130461092857782, 7.889390962021915453904115427372, 9.464139084238194490725727887405, 10.85431400831615286693931488093, 11.44246213003638864910147754517, 12.98391376410558553422219850277, 13.93567691836983128225658294599
