| L(s) = 1 | − i·2-s − 4-s + (1.73 + 1.41i)5-s + i·8-s + (1.41 − 1.73i)10-s − 2.82i·11-s + 16-s + (4 − 1.73i)19-s + (−1.73 − 1.41i)20-s − 2.82·22-s + 3.46·23-s + (0.999 + 4.89i)25-s − 2.44·29-s − 3.46i·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.774 + 0.632i)5-s + 0.353i·8-s + (0.447 − 0.547i)10-s − 0.852i·11-s + 0.250·16-s + (0.917 − 0.397i)19-s + (−0.387 − 0.316i)20-s − 0.603·22-s + 0.722·23-s + (0.199 + 0.979i)25-s − 0.454·29-s − 0.622i·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926999968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926999968\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.73 - 1.41i)T \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−9.304642322238897792485059123204, −8.744631692447241704189189215074, −7.61560454890817711098493484678, −6.83772315028441849710758493058, −5.78552453124397921012379973972, −5.26005859270998234391151908694, −3.94641528985926981944575460827, −3.06403028054718390479262643273, −2.26420185783552418714184506135, −0.938198945496714166866379820799,
1.08520405178667809610555877118, 2.32711201059392975004146712622, 3.72981323195686178378092898020, 4.77801729321754752691744696860, 5.39380764273165531097748043723, 6.14995435152013085085710548318, 7.15817008000338528270273107519, 7.67754170832535222066507277193, 8.922998276961206990475894170301, 9.112832973427977553981182213395
