| L(s) = 1 | + 0.618·3-s − 1.61·5-s − 0.618·7-s − 2.61·9-s + 0.854·13-s − 1.00·15-s + 4.85·17-s − 1.85·19-s − 0.381·21-s − 4·23-s − 2.38·25-s − 3.47·27-s − 8.32·29-s − 10.0·31-s + 1.00·35-s − 7.38·37-s + 0.527·39-s − 7.38·41-s + 10.4·43-s + 4.23·45-s + 9.56·47-s − 6.61·49-s + 3.00·51-s − 1.61·53-s − 1.14·57-s − 4.61·59-s + 4.85·61-s + ⋯ |
| L(s) = 1 | + 0.356·3-s − 0.723·5-s − 0.233·7-s − 0.872·9-s + 0.236·13-s − 0.258·15-s + 1.17·17-s − 0.425·19-s − 0.0833·21-s − 0.834·23-s − 0.476·25-s − 0.668·27-s − 1.54·29-s − 1.81·31-s + 0.169·35-s − 1.21·37-s + 0.0845·39-s − 1.15·41-s + 1.59·43-s + 0.631·45-s + 1.39·47-s − 0.945·49-s + 0.420·51-s − 0.222·53-s − 0.151·57-s − 0.601·59-s + 0.621·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 13 | \( 1 - 0.854T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 7.38T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 + 1.61T + 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 + 5.52T + 67T^{2} \) |
| 71 | \( 1 - 6.09T + 71T^{2} \) |
| 73 | \( 1 + 9.85T + 73T^{2} \) |
| 79 | \( 1 - 2.85T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 3.32T + 97T^{2} \) |
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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
−9.440641779066685401342939046575, −8.771811553898020880839308821769, −7.86307467986172443645176153073, −7.35278018492927500390542851351, −6.02829909448762209990088214931, −5.36704958470025533475891292204, −3.88477704108596484249207887343, −3.38218130462768167333156556197, −1.98237740972733751436662819945, 0,
1.98237740972733751436662819945, 3.38218130462768167333156556197, 3.88477704108596484249207887343, 5.36704958470025533475891292204, 6.02829909448762209990088214931, 7.35278018492927500390542851351, 7.86307467986172443645176153073, 8.771811553898020880839308821769, 9.440641779066685401342939046575
