| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s − 11-s − 12-s + 5.41·13-s − 2·15-s + 16-s + 2·17-s + 18-s − 2.24·19-s + 2·20-s − 22-s − 4.82·23-s − 24-s − 25-s + 5.41·26-s − 27-s + 9.65·29-s − 2·30-s + 6.24·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s + 1.50·13-s − 0.516·15-s + 0.250·16-s + 0.485·17-s + 0.235·18-s − 0.514·19-s + 0.447·20-s − 0.213·22-s − 1.00·23-s − 0.204·24-s − 0.200·25-s + 1.06·26-s − 0.192·27-s + 1.79·29-s − 0.365·30-s + 1.12·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.164966758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.164966758\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 - 5.41T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 9.65T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 0.828T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 + 7.89T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 5.07T + 83T^{2} \) |
| 89 | \( 1 - 3.75T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−8.452700009455326605632775384571, −7.981469552544518552284372504302, −6.73738043986355651898002442427, −6.17550687159741192834472047394, −5.80831539177917704294157620685, −4.85853113041344073102923490752, −4.11552285298945100391736118888, −3.11308842732284167142129486182, −2.07382726814357683375991202659, −1.05504638080848840359357740156,
1.05504638080848840359357740156, 2.07382726814357683375991202659, 3.11308842732284167142129486182, 4.11552285298945100391736118888, 4.85853113041344073102923490752, 5.80831539177917704294157620685, 6.17550687159741192834472047394, 6.73738043986355651898002442427, 7.981469552544518552284372504302, 8.452700009455326605632775384571
