| L(s) = 1 | − 1.21·2-s − 0.525·4-s − 2.90·7-s + 3.06·8-s − 0.214·11-s + 13-s + 3.52·14-s − 2.67·16-s + 6.42·17-s + 2.21·19-s + 0.260·22-s + 4.68·23-s − 1.21·26-s + 1.52·28-s − 8.70·29-s − 5.59·31-s − 2.88·32-s − 7.80·34-s − 2.28·37-s − 2.68·38-s − 3.05·41-s − 6.36·43-s + 0.112·44-s − 5.69·46-s + 1.09·47-s + 1.42·49-s − 0.525·52-s + ⋯ |
| L(s) = 1 | − 0.858·2-s − 0.262·4-s − 1.09·7-s + 1.08·8-s − 0.0646·11-s + 0.277·13-s + 0.942·14-s − 0.668·16-s + 1.55·17-s + 0.507·19-s + 0.0554·22-s + 0.977·23-s − 0.238·26-s + 0.288·28-s − 1.61·29-s − 1.00·31-s − 0.510·32-s − 1.33·34-s − 0.374·37-s − 0.436·38-s − 0.476·41-s − 0.970·43-s + 0.0169·44-s − 0.839·46-s + 0.159·47-s + 0.204·49-s − 0.0728·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7672694588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7672694588\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 + 0.214T + 11T^{2} \) |
| 17 | \( 1 - 6.42T + 17T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 + 2.28T + 37T^{2} \) |
| 41 | \( 1 + 3.05T + 41T^{2} \) |
| 43 | \( 1 + 6.36T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 + 6.23T + 53T^{2} \) |
| 59 | \( 1 - 9.26T + 59T^{2} \) |
| 61 | \( 1 + 0.280T + 61T^{2} \) |
| 67 | \( 1 - 7.76T + 67T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 9.52T + 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.902473796884561287865906102432, −8.063182358441160034383737025783, −7.39046298893166079966996198390, −6.73591409714256055949294835524, −5.62994317361755412195851127321, −5.06702810275798729442119800096, −3.72185011238245242632347857661, −3.27135036600373573874326845216, −1.76462944618003924273474079798, −0.62598591185290928969804972927,
0.62598591185290928969804972927, 1.76462944618003924273474079798, 3.27135036600373573874326845216, 3.72185011238245242632347857661, 5.06702810275798729442119800096, 5.62994317361755412195851127321, 6.73591409714256055949294835524, 7.39046298893166079966996198390, 8.063182358441160034383737025783, 8.902473796884561287865906102432
