| L(s) = 1 | − 2.84·5-s − 31.6·7-s − 11·11-s + 5.15·13-s − 121.·17-s − 34.8·19-s + 116.·23-s − 116.·25-s + 69.4·29-s − 140.·31-s + 90.3·35-s − 420.·37-s + 322.·41-s − 321.·43-s − 231.·47-s + 661.·49-s − 4.91·53-s + 31.3·55-s + 406.·59-s − 556.·61-s − 14.6·65-s − 84.7·67-s + 49.0·71-s + 785.·73-s + 348.·77-s + 383.·79-s − 930.·83-s + ⋯ |
| L(s) = 1 | − 0.254·5-s − 1.71·7-s − 0.301·11-s + 0.109·13-s − 1.73·17-s − 0.420·19-s + 1.05·23-s − 0.935·25-s + 0.444·29-s − 0.814·31-s + 0.436·35-s − 1.86·37-s + 1.22·41-s − 1.13·43-s − 0.718·47-s + 1.92·49-s − 0.0127·53-s + 0.0768·55-s + 0.896·59-s − 1.16·61-s − 0.0280·65-s − 0.154·67-s + 0.0820·71-s + 1.26·73-s + 0.516·77-s + 0.545·79-s − 1.23·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5846863216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5846863216\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 + 2.84T + 125T^{2} \) |
| 7 | \( 1 + 31.6T + 343T^{2} \) |
| 13 | \( 1 - 5.15T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 420.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 231.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 4.91T + 1.48e5T^{2} \) |
| 59 | \( 1 - 406.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 556.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 84.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 49.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 383.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 930.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 732.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−9.074627613738597635116263097065, −8.440469457646134598270649549990, −7.21311862253167329653135086711, −6.69855314238682753738554616271, −5.96901881740740176680070466932, −4.87209298601866010957361556333, −3.84732942619430744450117405086, −3.09118471652329952010141908045, −2.05466712005667014016377283090, −0.34606545875165740519834607431,
0.34606545875165740519834607431, 2.05466712005667014016377283090, 3.09118471652329952010141908045, 3.84732942619430744450117405086, 4.87209298601866010957361556333, 5.96901881740740176680070466932, 6.69855314238682753738554616271, 7.21311862253167329653135086711, 8.440469457646134598270649549990, 9.074627613738597635116263097065
