| L(s) = 1 | + 3-s + 2.77·5-s + 1.06·7-s + 9-s − 4.79·11-s + 1.09·13-s + 2.77·15-s − 4.90·17-s − 5.29·19-s + 1.06·21-s − 23-s + 2.68·25-s + 27-s − 29-s + 0.542·31-s − 4.79·33-s + 2.94·35-s − 9.49·37-s + 1.09·39-s − 0.882·41-s − 0.495·43-s + 2.77·45-s − 5.71·47-s − 5.87·49-s − 4.90·51-s − 0.364·53-s − 13.2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.23·5-s + 0.401·7-s + 0.333·9-s − 1.44·11-s + 0.303·13-s + 0.715·15-s − 1.19·17-s − 1.21·19-s + 0.231·21-s − 0.208·23-s + 0.536·25-s + 0.192·27-s − 0.185·29-s + 0.0974·31-s − 0.833·33-s + 0.497·35-s − 1.56·37-s + 0.175·39-s − 0.137·41-s − 0.0755·43-s + 0.413·45-s − 0.834·47-s − 0.838·49-s − 0.687·51-s − 0.0500·53-s − 1.79·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 2.77T + 5T^{2} \) |
| 7 | \( 1 - 1.06T + 7T^{2} \) |
| 11 | \( 1 + 4.79T + 11T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 17 | \( 1 + 4.90T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 31 | \( 1 - 0.542T + 31T^{2} \) |
| 37 | \( 1 + 9.49T + 37T^{2} \) |
| 41 | \( 1 + 0.882T + 41T^{2} \) |
| 43 | \( 1 + 0.495T + 43T^{2} \) |
| 47 | \( 1 + 5.71T + 47T^{2} \) |
| 53 | \( 1 + 0.364T + 53T^{2} \) |
| 59 | \( 1 + 6.03T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 + 4.59T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 7.11T + 89T^{2} \) |
| 97 | \( 1 - 2.03T + 97T^{2} \) |
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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
−7.60879063085310702606452918208, −6.66520737182000089991034239858, −6.21954909747126432546972273390, −5.27312618237434389802642957874, −4.83706386521698578411187633292, −3.88697876364257534007535351693, −2.86519426356174553047950390235, −2.14839595472672929861892983438, −1.68937674929665068575435042396, 0,
1.68937674929665068575435042396, 2.14839595472672929861892983438, 2.86519426356174553047950390235, 3.88697876364257534007535351693, 4.83706386521698578411187633292, 5.27312618237434389802642957874, 6.21954909747126432546972273390, 6.66520737182000089991034239858, 7.60879063085310702606452918208
