| L(s) = 1 | + 3-s + 5-s + 3.43·7-s + 9-s − 2.63·11-s − 13-s + 15-s + 7.84·17-s + 0.794·19-s + 3.43·21-s − 3.43·23-s + 25-s + 27-s − 4.06·29-s + 9.27·31-s − 2.63·33-s + 3.43·35-s − 0.636·37-s − 39-s − 4.63·41-s + 2.41·43-s + 45-s + 5.27·47-s + 4.77·49-s + 7.84·51-s + 11.4·53-s − 2.63·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.29·7-s + 0.333·9-s − 0.795·11-s − 0.277·13-s + 0.258·15-s + 1.90·17-s + 0.182·19-s + 0.748·21-s − 0.715·23-s + 0.200·25-s + 0.192·27-s − 0.755·29-s + 1.66·31-s − 0.458·33-s + 0.580·35-s − 0.104·37-s − 0.160·39-s − 0.724·41-s + 0.367·43-s + 0.149·45-s + 0.769·47-s + 0.682·49-s + 1.09·51-s + 1.57·53-s − 0.355·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.658191725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.658191725\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 17 | \( 1 - 7.84T + 17T^{2} \) |
| 19 | \( 1 - 0.794T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 - 9.27T + 31T^{2} \) |
| 37 | \( 1 + 0.636T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 - 2.41T + 43T^{2} \) |
| 47 | \( 1 - 5.27T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 6.41T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 8.63T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−9.472317953188781602547058671510, −8.488176953244069843852140818610, −7.83284790028976280581249569926, −7.41644598289367513556424409482, −6.02078879345166859751202156638, −5.27280077978339403064666770068, −4.50018410272704688600700437853, −3.29129373650755031170999303490, −2.29421508134718927524124795597, −1.26170598600573382275781644085,
1.26170598600573382275781644085, 2.29421508134718927524124795597, 3.29129373650755031170999303490, 4.50018410272704688600700437853, 5.27280077978339403064666770068, 6.02078879345166859751202156638, 7.41644598289367513556424409482, 7.83284790028976280581249569926, 8.488176953244069843852140818610, 9.472317953188781602547058671510
