| L(s) = 1 | + 2.08i·3-s − i·5-s − 1.84·7-s − 1.32·9-s + 3.64·11-s − 3.35·13-s + 2.08·15-s + 4.09i·17-s + 2.96·19-s − 3.84i·21-s + (−1.07 − 4.67i)23-s − 25-s + 3.47i·27-s − 3.51·29-s − 2.72i·31-s + ⋯ |
| L(s) = 1 | + 1.20i·3-s − 0.447i·5-s − 0.698·7-s − 0.442·9-s + 1.09·11-s − 0.930·13-s + 0.537·15-s + 0.994i·17-s + 0.680·19-s − 0.839i·21-s + (−0.224 − 0.974i)23-s − 0.200·25-s + 0.669i·27-s − 0.652·29-s − 0.490i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.224408119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224408119\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (1.07 + 4.67i)T \) |
| good | 3 | \( 1 - 2.08iT - 3T^{2} \) |
| 7 | \( 1 + 1.84T + 7T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 + 3.35T + 13T^{2} \) |
| 17 | \( 1 - 4.09iT - 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 29 | \( 1 + 3.51T + 29T^{2} \) |
| 31 | \( 1 + 2.72iT - 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 9.90T + 41T^{2} \) |
| 43 | \( 1 - 2.15T + 43T^{2} \) |
| 47 | \( 1 - 3.74iT - 47T^{2} \) |
| 53 | \( 1 - 2.18iT - 53T^{2} \) |
| 59 | \( 1 + 7.63iT - 59T^{2} \) |
| 61 | \( 1 - 6.81iT - 61T^{2} \) |
| 67 | \( 1 - 0.877T + 67T^{2} \) |
| 71 | \( 1 - 8.98iT - 71T^{2} \) |
| 73 | \( 1 + 6.61T + 73T^{2} \) |
| 79 | \( 1 + 7.60T + 79T^{2} \) |
| 83 | \( 1 - 2.97T + 83T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 10.5iT - 97T^{2} \) |
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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
−9.046239294619256719251470005151, −8.291116472892664382305210683357, −7.36114225160315261532334976156, −6.48171390305060979654864015787, −5.80387883052571975314396856469, −4.84704412226219232680033322188, −4.23193887550771255311070696723, −3.62511710115630341789169463018, −2.63014273033735649581491750609, −1.24613181249954812065049085245,
0.38057683653578515781956822187, 1.56368436872639630228347275179, 2.49448019626624961370872976474, 3.35821284255946286754425177862, 4.28671620743805238117560297888, 5.49035716396647367738438177582, 6.13585424302164028959753834864, 7.06124708876157454716951047877, 7.21763252180318731593897672425, 7.88825201564483939792635853129
