| L(s) = 1 | − 2-s + 4-s + (−1.85 + 1.25i)5-s − 4.78i·7-s − 8-s + (1.85 − 1.25i)10-s + 5.98·11-s + 3.49·13-s + 4.78i·14-s + 16-s + 4.96·17-s + 7.33i·19-s + (−1.85 + 1.25i)20-s − 5.98·22-s − 1.74·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.829 + 0.559i)5-s − 1.81i·7-s − 0.353·8-s + (0.586 − 0.395i)10-s + 1.80·11-s + 0.969·13-s + 1.28i·14-s + 0.250·16-s + 1.20·17-s + 1.68i·19-s + (−0.414 + 0.279i)20-s − 1.27·22-s − 0.364·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.404583852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.404583852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.85 - 1.25i)T \) |
| 37 | \( 1 + (-3.96 - 4.61i)T \) |
| good | 7 | \( 1 + 4.78iT - 7T^{2} \) |
| 11 | \( 1 - 5.98T + 11T^{2} \) |
| 13 | \( 1 - 3.49T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 - 7.33iT - 19T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 + 7.85iT - 29T^{2} \) |
| 31 | \( 1 - 3.24iT - 31T^{2} \) |
| 41 | \( 1 - 0.530T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 - 4.30iT - 47T^{2} \) |
| 53 | \( 1 - 3.66iT - 53T^{2} \) |
| 59 | \( 1 - 2.15iT - 59T^{2} \) |
| 61 | \( 1 + 3.06iT - 61T^{2} \) |
| 67 | \( 1 + 3.79iT - 67T^{2} \) |
| 71 | \( 1 - 8.47T + 71T^{2} \) |
| 73 | \( 1 - 9.05iT - 73T^{2} \) |
| 79 | \( 1 + 5.56iT - 79T^{2} \) |
| 83 | \( 1 - 3.77iT - 83T^{2} \) |
| 89 | \( 1 - 8.45iT - 89T^{2} \) |
| 97 | \( 1 + 3.64T + 97T^{2} \) |
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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
−8.248160959558658954918805545928, −7.976611338083214990743320267182, −7.22352675371249181086419762533, −6.51795174168398301529848077347, −6.00573641435117502092484040699, −4.29943751057262953828834759733, −3.82474227454335416875610890994, −3.30884882048905860551910271495, −1.48952158937266780552419977094, −0.863087806615903973487493483635,
0.855860366264132291587435011298, 1.84362357541839188186203692782, 3.06688513987578601965084844706, 3.81378799932589682920231978212, 4.93802823268225033261033990995, 5.74674162985441109488358384881, 6.46696691800208320426736964535, 7.25229825590543431487490679881, 8.220228909771189132376821364707, 8.819094551623654831697847414291
