| L(s) = 1 | − 2-s + 4-s − 5-s + 3.35·7-s − 8-s + 10-s + 3.76·11-s − 2.80·13-s − 3.35·14-s + 16-s − 3.35·17-s − 4.31·19-s − 20-s − 3.76·22-s − 2.96·23-s + 25-s + 2.80·26-s + 3.35·28-s − 10.0·29-s + 31-s − 32-s + 3.35·34-s − 3.35·35-s − 9.50·37-s + 4.31·38-s + 40-s + 11.2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.26·7-s − 0.353·8-s + 0.316·10-s + 1.13·11-s − 0.778·13-s − 0.895·14-s + 0.250·16-s − 0.812·17-s − 0.989·19-s − 0.223·20-s − 0.803·22-s − 0.617·23-s + 0.200·25-s + 0.550·26-s + 0.633·28-s − 1.87·29-s + 0.179·31-s − 0.176·32-s + 0.574·34-s − 0.566·35-s − 1.56·37-s + 0.699·38-s + 0.158·40-s + 1.76·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 - 3.76T + 11T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 + 4.31T + 19T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 37 | \( 1 + 9.50T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 9.89T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 6.38T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 - 3.66T + 73T^{2} \) |
| 79 | \( 1 + 0.836T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 - 0.700T + 89T^{2} \) |
| 97 | \( 1 + 0.261T + 97T^{2} \) |
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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
−8.480217133865551076501113604269, −7.77049702516417610593242163121, −7.08319779566413917021616772792, −6.36297011182004459593444161911, −5.28925114815133409173170641293, −4.42123178336330580689391687160, −3.69163654852904855886714140522, −2.22840562583334235370799844419, −1.56354460748463021456872588694, 0,
1.56354460748463021456872588694, 2.22840562583334235370799844419, 3.69163654852904855886714140522, 4.42123178336330580689391687160, 5.28925114815133409173170641293, 6.36297011182004459593444161911, 7.08319779566413917021616772792, 7.77049702516417610593242163121, 8.480217133865551076501113604269
