| L(s) = 1 | − 3-s − 2.82·5-s − 2.82·7-s + 9-s + 2·11-s + 13-s + 2.82·15-s − 3.65·17-s − 2.82·19-s + 2.82·21-s − 4·23-s + 3.00·25-s − 27-s − 2·29-s − 6.82·31-s − 2·33-s + 8.00·35-s − 3.65·37-s − 39-s + 10.8·41-s − 9.65·43-s − 2.82·45-s − 0.343·47-s + 1.00·49-s + 3.65·51-s + 2·53-s − 5.65·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.26·5-s − 1.06·7-s + 0.333·9-s + 0.603·11-s + 0.277·13-s + 0.730·15-s − 0.886·17-s − 0.648·19-s + 0.617·21-s − 0.834·23-s + 0.600·25-s − 0.192·27-s − 0.371·29-s − 1.22·31-s − 0.348·33-s + 1.35·35-s − 0.601·37-s − 0.160·39-s + 1.69·41-s − 1.47·43-s − 0.421·45-s − 0.0500·47-s + 0.142·49-s + 0.512·51-s + 0.274·53-s − 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5378928251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5378928251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−8.927087577014521518136321072367, −8.130248255289238952859635806188, −7.26707826261278552196695869442, −6.60967813022995481805985243685, −6.01047093862231415893175612814, −4.88661667460457854686950327026, −3.88042272842888373487727079448, −3.60545430611737042969241687746, −2.10756985928360306791363446009, −0.45849006859332749366239737346,
0.45849006859332749366239737346, 2.10756985928360306791363446009, 3.60545430611737042969241687746, 3.88042272842888373487727079448, 4.88661667460457854686950327026, 6.01047093862231415893175612814, 6.60967813022995481805985243685, 7.26707826261278552196695869442, 8.130248255289238952859635806188, 8.927087577014521518136321072367
