| L(s) = 1 | + 3-s + 2.12·7-s + 9-s + 3.15·11-s − 13-s − 0.515·17-s + 6.73·19-s + 2.12·21-s + 4.24·23-s + 27-s + 0.0302·29-s + 7.15·31-s + 3.15·33-s − 5.76·37-s − 39-s − 3.76·41-s + 3.28·43-s + 4.60·47-s − 2.48·49-s − 0.515·51-s − 0.280·53-s + 6.73·57-s + 11.3·59-s − 5.01·61-s + 2.12·63-s − 15.1·67-s + 4.24·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.803·7-s + 0.333·9-s + 0.951·11-s − 0.277·13-s − 0.124·17-s + 1.54·19-s + 0.463·21-s + 0.886·23-s + 0.192·27-s + 0.00562·29-s + 1.28·31-s + 0.549·33-s − 0.947·37-s − 0.160·39-s − 0.587·41-s + 0.500·43-s + 0.672·47-s − 0.354·49-s − 0.0721·51-s − 0.0384·53-s + 0.892·57-s + 1.47·59-s − 0.642·61-s + 0.267·63-s − 1.84·67-s + 0.511·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.432342315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.432342315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 17 | \( 1 + 0.515T + 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 - 0.0302T + 29T^{2} \) |
| 31 | \( 1 - 7.15T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 + 3.76T + 41T^{2} \) |
| 43 | \( 1 - 3.28T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + 0.280T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 5.01T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 2.23T + 71T^{2} \) |
| 73 | \( 1 - 2.96T + 73T^{2} \) |
| 79 | \( 1 + 8.98T + 79T^{2} \) |
| 83 | \( 1 + 5.40T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 0.909T + 97T^{2} \) |
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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
−7.79143995920375914093072240097, −7.27187203208441415433761563586, −6.64830092603914128731069642449, −5.69828887806746847553125071080, −4.93953817050152635471610312513, −4.34253535925728150234271408013, −3.43796396340699461878803761398, −2.76464827656892912960537622508, −1.69822645751751309983858918802, −0.980656175413720460102212646079,
0.980656175413720460102212646079, 1.69822645751751309983858918802, 2.76464827656892912960537622508, 3.43796396340699461878803761398, 4.34253535925728150234271408013, 4.93953817050152635471610312513, 5.69828887806746847553125071080, 6.64830092603914128731069642449, 7.27187203208441415433761563586, 7.79143995920375914093072240097
