L(s) = 1 | + (−64 + 64i)2-s + (−1.76e3 − 1.76e3i)3-s − 8.19e3i·4-s + (4.62e4 − 6.29e4i)5-s + 2.25e5·6-s + (−5.96e4 + 5.96e4i)7-s + (5.24e5 + 5.24e5i)8-s + 1.41e6i·9-s + (1.07e6 + 6.98e6i)10-s − 4.64e6·11-s + (−1.44e7 + 1.44e7i)12-s + (−4.85e7 − 4.85e7i)13-s − 7.63e6i·14-s + (−1.92e8 + 2.95e7i)15-s − 6.71e7·16-s + (−4.70e8 + 4.70e8i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (−0.805 − 0.805i)3-s − 0.5i·4-s + (0.591 − 0.806i)5-s + 0.805·6-s + (−0.0724 + 0.0724i)7-s + (0.250 + 0.250i)8-s + 0.296i·9-s + (0.107 + 0.698i)10-s − 0.238·11-s + (−0.402 + 0.402i)12-s + (−0.773 − 0.773i)13-s − 0.0724i·14-s + (−1.12 + 0.172i)15-s − 0.250·16-s + (−1.14 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(0.0253527 + 0.124646i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0253527 + 0.124646i\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (64 - 64i)T \) |
| 5 | \( 1 + (-4.62e4 + 6.29e4i)T \) |
good | 3 | \( 1 + (1.76e3 + 1.76e3i)T + 4.78e6iT^{2} \) |
| 7 | \( 1 + (5.96e4 - 5.96e4i)T - 6.78e11iT^{2} \) |
| 11 | \( 1 + 4.64e6T + 3.79e14T^{2} \) |
| 13 | \( 1 + (4.85e7 + 4.85e7i)T + 3.93e15iT^{2} \) |
| 17 | \( 1 + (4.70e8 - 4.70e8i)T - 1.68e17iT^{2} \) |
| 19 | \( 1 - 7.64e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 + (-1.43e9 - 1.43e9i)T + 1.15e19iT^{2} \) |
| 29 | \( 1 - 3.55e9iT - 2.97e20T^{2} \) |
| 31 | \( 1 - 4.07e10T + 7.56e20T^{2} \) |
| 37 | \( 1 + (1.25e11 - 1.25e11i)T - 9.01e21iT^{2} \) |
| 41 | \( 1 + 1.93e11T + 3.79e22T^{2} \) |
| 43 | \( 1 + (2.99e11 + 2.99e11i)T + 7.38e22iT^{2} \) |
| 47 | \( 1 + (-4.49e11 + 4.49e11i)T - 2.56e23iT^{2} \) |
| 53 | \( 1 + (8.73e11 + 8.73e11i)T + 1.37e24iT^{2} \) |
| 59 | \( 1 + 2.30e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 1.81e12T + 9.87e24T^{2} \) |
| 67 | \( 1 + (5.96e12 - 5.96e12i)T - 3.67e25iT^{2} \) |
| 71 | \( 1 - 5.96e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + (7.19e11 + 7.19e11i)T + 1.22e26iT^{2} \) |
| 79 | \( 1 + 3.29e12iT - 3.68e26T^{2} \) |
| 83 | \( 1 + (6.80e12 + 6.80e12i)T + 7.36e26iT^{2} \) |
| 89 | \( 1 + 3.40e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (7.87e13 - 7.87e13i)T - 6.52e27iT^{2} \) |
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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
−17.11140427063895259462304083769, −15.36195143466284017641259753739, −13.34276562414128841903193687214, −12.15982491619917716643005231494, −10.18654748253584399714000290982, −8.394487767579140120516758800473, −6.59106126734779167303545087104, −5.29408584535899818843583703261, −1.56967147870654896428590433510, −0.07033703289042689368084490191,
2.49849083420840402233099454941, 4.77120784739518636035285519998, 6.85402815199494331227929449361, 9.397543256642367626990247575327, 10.56461978604801446536959793434, 11.59717014289871328957556814619, 13.69636619831841176957843840594, 15.57653927272628597691951429369, 16.96506739253941094161451927645, 17.96570195901098597583292085136