| L(s) = 1 | − 1.44·3-s + 5-s − 2.92·7-s − 0.917·9-s − 4.55·11-s + 1.15·13-s − 1.44·15-s + 5.35·17-s + 19-s + 4.22·21-s − 6.10·23-s + 25-s + 5.65·27-s + 5.09·29-s + 1.54·31-s + 6.57·33-s − 2.92·35-s + 8.92·37-s − 1.66·39-s − 2.88·41-s + 7.86·43-s − 0.917·45-s + 7.58·47-s + 1.55·49-s − 7.73·51-s − 4.48·53-s − 4.55·55-s + ⋯ |
| L(s) = 1 | − 0.833·3-s + 0.447·5-s − 1.10·7-s − 0.305·9-s − 1.37·11-s + 0.320·13-s − 0.372·15-s + 1.29·17-s + 0.229·19-s + 0.921·21-s − 1.27·23-s + 0.200·25-s + 1.08·27-s + 0.945·29-s + 0.276·31-s + 1.14·33-s − 0.494·35-s + 1.46·37-s − 0.266·39-s − 0.450·41-s + 1.19·43-s − 0.136·45-s + 1.10·47-s + 0.222·49-s − 1.08·51-s − 0.615·53-s − 0.614·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.44T + 3T^{2} \) |
| 7 | \( 1 + 2.92T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 - 5.35T + 17T^{2} \) |
| 23 | \( 1 + 6.10T + 23T^{2} \) |
| 29 | \( 1 - 5.09T + 29T^{2} \) |
| 31 | \( 1 - 1.54T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 + 2.88T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 - 2.09T + 59T^{2} \) |
| 61 | \( 1 + 1.59T + 61T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 + 9.81T + 73T^{2} \) |
| 79 | \( 1 + 4.60T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 - 1.91T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−7.79363601827299196102968721349, −6.82180007204278855254192920525, −6.03155549778138934193702723083, −5.78143501734353911520900110531, −5.08165613359732927637566363841, −4.09110844312968179117005902185, −3.01718865841320761580297928580, −2.56047966093840703856909768037, −1.04657780811973872249637253920, 0,
1.04657780811973872249637253920, 2.56047966093840703856909768037, 3.01718865841320761580297928580, 4.09110844312968179117005902185, 5.08165613359732927637566363841, 5.78143501734353911520900110531, 6.03155549778138934193702723083, 6.82180007204278855254192920525, 7.79363601827299196102968721349
