| L(s) = 1 | + 3-s − 1.98·5-s − 0.0909·7-s + 9-s − 2.35·11-s + 0.834·13-s − 1.98·15-s + 7.08·17-s − 1.66·19-s − 0.0909·21-s − 2.49·23-s − 1.05·25-s + 27-s − 3.96·29-s − 3.52·31-s − 2.35·33-s + 0.180·35-s − 1.64·37-s + 0.834·39-s + 0.142·41-s − 5.36·43-s − 1.98·45-s − 6.35·47-s − 6.99·49-s + 7.08·51-s + 13.4·53-s + 4.67·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.888·5-s − 0.0343·7-s + 0.333·9-s − 0.710·11-s + 0.231·13-s − 0.512·15-s + 1.71·17-s − 0.382·19-s − 0.0198·21-s − 0.521·23-s − 0.211·25-s + 0.192·27-s − 0.736·29-s − 0.633·31-s − 0.410·33-s + 0.0305·35-s − 0.270·37-s + 0.133·39-s + 0.0223·41-s − 0.817·43-s − 0.296·45-s − 0.926·47-s − 0.998·49-s + 0.992·51-s + 1.84·53-s + 0.630·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 1.98T + 5T^{2} \) |
| 7 | \( 1 + 0.0909T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 0.834T + 13T^{2} \) |
| 17 | \( 1 - 7.08T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 + 2.49T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 + 3.52T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 - 0.142T + 41T^{2} \) |
| 43 | \( 1 + 5.36T + 43T^{2} \) |
| 47 | \( 1 + 6.35T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 4.39T + 59T^{2} \) |
| 61 | \( 1 - 0.0766T + 61T^{2} \) |
| 67 | \( 1 - 4.51T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 - 9.30T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 - 0.0141T + 83T^{2} \) |
| 89 | \( 1 + 7.96T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.109158409378309537396991928880, −7.54126955541304936273479435675, −6.83950226391713457094170382330, −5.73385309615965575104891251543, −5.11272114179857448220479912907, −3.95407517375463636507934667037, −3.55678940592413695348604713581, −2.60225418646416617478139066579, −1.46454003648064016646555818374, 0,
1.46454003648064016646555818374, 2.60225418646416617478139066579, 3.55678940592413695348604713581, 3.95407517375463636507934667037, 5.11272114179857448220479912907, 5.73385309615965575104891251543, 6.83950226391713457094170382330, 7.54126955541304936273479435675, 8.109158409378309537396991928880
