| L(s) = 1 | + (0.951 + 0.309i)2-s + (−1.58 + 2.17i)3-s + (0.809 + 0.587i)4-s + (0.479 + 2.18i)5-s + (−2.17 + 1.58i)6-s + i·7-s + (0.587 + 0.809i)8-s + (−1.30 − 4.02i)9-s + (−0.218 + 2.22i)10-s + (0.920 − 2.83i)11-s + (−2.55 + 0.831i)12-s + (−3.71 + 1.20i)13-s + (−0.309 + 0.951i)14-s + (−5.51 − 2.40i)15-s + (0.309 + 0.951i)16-s + (2.04 + 2.81i)17-s + ⋯ |
| L(s) = 1 | + (0.672 + 0.218i)2-s + (−0.912 + 1.25i)3-s + (0.404 + 0.293i)4-s + (0.214 + 0.976i)5-s + (−0.888 + 0.645i)6-s + 0.377i·7-s + (0.207 + 0.286i)8-s + (−0.436 − 1.34i)9-s + (−0.0691 + 0.703i)10-s + (0.277 − 0.854i)11-s + (−0.738 + 0.239i)12-s + (−1.03 + 0.335i)13-s + (−0.0825 + 0.254i)14-s + (−1.42 − 0.622i)15-s + (0.0772 + 0.237i)16-s + (0.496 + 0.683i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.395638 + 1.34293i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.395638 + 1.34293i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (-0.479 - 2.18i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (1.58 - 2.17i)T + (-0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (-0.920 + 2.83i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (3.71 - 1.20i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.04 - 2.81i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.04 + 2.93i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (5.97 + 1.94i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.40 - 3.92i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.39 - 1.01i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-8.43 + 2.74i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.32 - 10.2i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 9.22iT - 43T^{2} \) |
| 47 | \( 1 + (-4.13 + 5.69i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.70 + 6.47i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.12 + 9.60i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.513 + 1.58i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-5.51 - 7.58i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (4.51 + 3.27i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-7.11 - 2.31i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.25 - 1.64i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.92 - 2.64i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.829 - 2.55i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (5.86 - 8.07i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−11.57665742581689596752503771509, −11.14351197674204369218339641234, −10.12792235081557513429961706150, −9.502758809681372083085276620293, −7.987074035484372185926680545686, −6.63067693395021303812811186578, −5.90394345999387760076444455963, −5.01955913748790825992072606417, −3.91826110127213175336023842172, −2.79401974526277456851856337846,
0.905635102972393195532249099920, 2.20570793525832248052414363531, 4.25588754753670673922009291056, 5.33239236819279564820387897804, 6.01469785288540688003859711481, 7.30810588250753348970294196525, 7.75708408420060934987483100712, 9.508755714041270826757597342633, 10.28800402960744142241522434044, 11.74543660065172466510555148585
