| L(s) = 1 | − 1.09·2-s + 3-s − 0.803·4-s − 3.36·5-s − 1.09·6-s + 7-s + 3.06·8-s + 9-s + 3.68·10-s − 0.408·11-s − 0.803·12-s − 1.09·14-s − 3.36·15-s − 1.74·16-s + 0.412·17-s − 1.09·18-s + 1.06·19-s + 2.70·20-s + 21-s + 0.446·22-s + 8.83·23-s + 3.06·24-s + 6.35·25-s + 27-s − 0.803·28-s − 3.54·29-s + 3.68·30-s + ⋯ |
| L(s) = 1 | − 0.773·2-s + 0.577·3-s − 0.401·4-s − 1.50·5-s − 0.446·6-s + 0.377·7-s + 1.08·8-s + 0.333·9-s + 1.16·10-s − 0.123·11-s − 0.232·12-s − 0.292·14-s − 0.870·15-s − 0.436·16-s + 0.0999·17-s − 0.257·18-s + 0.244·19-s + 0.605·20-s + 0.218·21-s + 0.0951·22-s + 1.84·23-s + 0.625·24-s + 1.27·25-s + 0.192·27-s − 0.151·28-s − 0.658·29-s + 0.672·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8544557898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8544557898\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.09T + 2T^{2} \) |
| 5 | \( 1 + 3.36T + 5T^{2} \) |
| 11 | \( 1 + 0.408T + 11T^{2} \) |
| 17 | \( 1 - 0.412T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 - 8.83T + 23T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 31 | \( 1 + 1.89T + 31T^{2} \) |
| 37 | \( 1 + 4.79T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 + 8.84T + 47T^{2} \) |
| 53 | \( 1 + 4.18T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 4.27T + 79T^{2} \) |
| 83 | \( 1 - 4.60T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.509269847940840114916639743738, −8.038198936468281419317082666098, −7.30802638853541903998160545395, −6.89046097027904212387851701513, −5.21126554087876532663835834756, −4.73200116838652122040183881686, −3.76047110286007575511409789509, −3.22480127970824104403316939927, −1.75412559749040798485289777308, −0.61233151487875881174815423061,
0.61233151487875881174815423061, 1.75412559749040798485289777308, 3.22480127970824104403316939927, 3.76047110286007575511409789509, 4.73200116838652122040183881686, 5.21126554087876532663835834756, 6.89046097027904212387851701513, 7.30802638853541903998160545395, 8.038198936468281419317082666098, 8.509269847940840114916639743738
