| L(s) = 1 | − 0.598·3-s − 0.722·5-s − 0.897·7-s − 2.64·9-s + 3.15·11-s − 5.02·13-s + 0.432·15-s + 17-s + 5.60·19-s + 0.536·21-s + 6.04·23-s − 4.47·25-s + 3.37·27-s + 1.54·29-s − 0.336·31-s − 1.88·33-s + 0.648·35-s + 4.94·37-s + 3.00·39-s + 7.50·41-s − 7.93·43-s + 1.90·45-s + 5.10·47-s − 6.19·49-s − 0.598·51-s − 9.86·53-s − 2.27·55-s + ⋯ |
| L(s) = 1 | − 0.345·3-s − 0.323·5-s − 0.339·7-s − 0.880·9-s + 0.951·11-s − 1.39·13-s + 0.111·15-s + 0.242·17-s + 1.28·19-s + 0.117·21-s + 1.26·23-s − 0.895·25-s + 0.649·27-s + 0.286·29-s − 0.0604·31-s − 0.328·33-s + 0.109·35-s + 0.813·37-s + 0.481·39-s + 1.17·41-s − 1.21·43-s + 0.284·45-s + 0.743·47-s − 0.884·49-s − 0.0837·51-s − 1.35·53-s − 0.307·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 0.598T + 3T^{2} \) |
| 5 | \( 1 + 0.722T + 5T^{2} \) |
| 7 | \( 1 + 0.897T + 7T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 13 | \( 1 + 5.02T + 13T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 - 1.54T + 29T^{2} \) |
| 31 | \( 1 + 0.336T + 31T^{2} \) |
| 37 | \( 1 - 4.94T + 37T^{2} \) |
| 41 | \( 1 - 7.50T + 41T^{2} \) |
| 43 | \( 1 + 7.93T + 43T^{2} \) |
| 47 | \( 1 - 5.10T + 47T^{2} \) |
| 53 | \( 1 + 9.86T + 53T^{2} \) |
| 61 | \( 1 + 2.08T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 + 0.378T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 2.25T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 6.09T + 89T^{2} \) |
| 97 | \( 1 + 2.30T + 97T^{2} \) |
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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
−7.951037198269179146390671118492, −7.35805492233789715347424263402, −6.62558037042613860644755583205, −5.83135368712286251049205715874, −5.12267498155684431009150716256, −4.35607078004761317085208313961, −3.27861698093084442326136434120, −2.68794255984127279559351724223, −1.24790195271415593322700143825, 0,
1.24790195271415593322700143825, 2.68794255984127279559351724223, 3.27861698093084442326136434120, 4.35607078004761317085208313961, 5.12267498155684431009150716256, 5.83135368712286251049205715874, 6.62558037042613860644755583205, 7.35805492233789715347424263402, 7.951037198269179146390671118492
