| L(s) = 1 | + (1 − i)2-s + (1.58 − 1.58i)5-s + (−0.581 − 2.58i)7-s + (2 + 2i)8-s − 3.16i·10-s + 11-s + (−1.58 + 1.58i)13-s + (−3.16 − 2i)14-s + 4·16-s + (−1.58 − 1.58i)17-s + 3.16·19-s + (1 − i)22-s + (−2 − 2i)23-s − 5.00i·25-s + 3.16i·26-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)5-s + (−0.219 − 0.975i)7-s + (0.707 + 0.707i)8-s − 1.00i·10-s + 0.301·11-s + (−0.438 + 0.438i)13-s + (−0.845 − 0.534i)14-s + 16-s + (−0.383 − 0.383i)17-s + 0.725·19-s + (0.213 − 0.213i)22-s + (−0.417 − 0.417i)23-s − 1.00i·25-s + 0.620i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74454 - 1.09072i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74454 - 1.09072i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.58 + 1.58i)T \) |
| 7 | \( 1 + (0.581 + 2.58i)T \) |
| good | 2 | \( 1 + (-1 + i)T - 2iT^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + (1.58 - 1.58i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.58 + 1.58i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + (2 + 2i)T + 23iT^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 - 3.16iT - 31T^{2} \) |
| 37 | \( 1 + (6 - 6i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.48iT - 41T^{2} \) |
| 43 | \( 1 + (3 + 3i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.74 - 4.74i)T + 47iT^{2} \) |
| 53 | \( 1 + (1 + i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 - 6.32iT - 61T^{2} \) |
| 67 | \( 1 + (1 - i)T - 67iT^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 + 13iT - 79T^{2} \) |
| 83 | \( 1 + (-3.16 + 3.16i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 + (1.58 + 1.58i)T + 97iT^{2} \) |
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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
−11.73842403705828335071219904378, −10.65385489279237961667549441439, −9.836763660604467706312328341122, −8.807421473737900457411843848123, −7.63339777848604071473896654626, −6.52690765283021155733060995138, −5.06324056398501151435485557082, −4.34831625014759598932213254230, −3.06583101621596069413765010257, −1.55090239595375865639954210336,
2.13191334659042768959220316744, 3.63016684718638810785968938173, 5.20694083108049077105125199954, 5.85770381043851349133071457343, 6.68631560571635321228777315088, 7.68130652035388345136863093330, 9.149981013027140106365553654337, 9.918459669709633565628360605224, 10.82519585479477571320196020160, 12.01739035434676919289925875985
