| L(s) = 1 | − 1.31·2-s − 1.97·3-s − 0.272·4-s − 1.38·5-s + 2.59·6-s − 2.03·7-s + 2.98·8-s + 0.908·9-s + 1.82·10-s + 0.539·12-s − 5.73·13-s + 2.66·14-s + 2.74·15-s − 3.37·16-s − 6.94·17-s − 1.19·18-s + 3.55·19-s + 0.378·20-s + 4.01·21-s + 0.806·23-s − 5.90·24-s − 3.07·25-s + 7.54·26-s + 4.13·27-s + 0.553·28-s − 8.32·29-s − 3.60·30-s + ⋯ |
| L(s) = 1 | − 0.929·2-s − 1.14·3-s − 0.136·4-s − 0.620·5-s + 1.06·6-s − 0.767·7-s + 1.05·8-s + 0.302·9-s + 0.576·10-s + 0.155·12-s − 1.59·13-s + 0.713·14-s + 0.708·15-s − 0.844·16-s − 1.68·17-s − 0.281·18-s + 0.814·19-s + 0.0846·20-s + 0.875·21-s + 0.168·23-s − 1.20·24-s − 0.614·25-s + 1.47·26-s + 0.795·27-s + 0.104·28-s − 1.54·29-s − 0.658·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 3 | \( 1 + 1.97T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 + 2.03T + 7T^{2} \) |
| 13 | \( 1 + 5.73T + 13T^{2} \) |
| 17 | \( 1 + 6.94T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 - 0.806T + 23T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 - 4.04T + 41T^{2} \) |
| 43 | \( 1 + 0.832T + 43T^{2} \) |
| 47 | \( 1 - 4.66T + 47T^{2} \) |
| 53 | \( 1 + 9.49T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 67 | \( 1 - 6.13T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 8.32T + 73T^{2} \) |
| 79 | \( 1 + 3.78T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 7.09T + 97T^{2} \) |
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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
−7.56849272427596709100354878871, −6.99062765604314540448945829011, −6.28829849535177379789137956022, −5.47322640968721304296943970469, −4.65389002408396975275766437619, −4.26530631790093802699380724225, −3.04911177725296319687851750775, −2.03340039513257286064691075032, −0.61850641907896478419357944024, 0,
0.61850641907896478419357944024, 2.03340039513257286064691075032, 3.04911177725296319687851750775, 4.26530631790093802699380724225, 4.65389002408396975275766437619, 5.47322640968721304296943970469, 6.28829849535177379789137956022, 6.99062765604314540448945829011, 7.56849272427596709100354878871
