| L(s) = 1 | − 0.291·2-s + 3-s − 1.91·4-s + 3.62·5-s − 0.291·6-s + 1.35·7-s + 1.14·8-s + 9-s − 1.05·10-s + 11-s − 1.91·12-s − 0.394·14-s + 3.62·15-s + 3.49·16-s − 3.00·17-s − 0.291·18-s − 7.12·19-s − 6.94·20-s + 1.35·21-s − 0.291·22-s − 1.04·23-s + 1.14·24-s + 8.13·25-s + 27-s − 2.59·28-s + 8.54·29-s − 1.05·30-s + ⋯ |
| L(s) = 1 | − 0.206·2-s + 0.577·3-s − 0.957·4-s + 1.62·5-s − 0.119·6-s + 0.511·7-s + 0.403·8-s + 0.333·9-s − 0.334·10-s + 0.301·11-s − 0.552·12-s − 0.105·14-s + 0.935·15-s + 0.874·16-s − 0.728·17-s − 0.0687·18-s − 1.63·19-s − 1.55·20-s + 0.295·21-s − 0.0621·22-s − 0.218·23-s + 0.232·24-s + 1.62·25-s + 0.192·27-s − 0.489·28-s + 1.58·29-s − 0.192·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.660162660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.660162660\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.291T + 2T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 - 8.54T + 29T^{2} \) |
| 31 | \( 1 + 6.40T + 31T^{2} \) |
| 37 | \( 1 - 9.77T + 37T^{2} \) |
| 41 | \( 1 + 2.49T + 41T^{2} \) |
| 43 | \( 1 - 9.35T + 43T^{2} \) |
| 47 | \( 1 - 6.55T + 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 + 1.88T + 59T^{2} \) |
| 61 | \( 1 - 6.08T + 61T^{2} \) |
| 67 | \( 1 - 8.83T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 - 7.17T + 79T^{2} \) |
| 83 | \( 1 - 2.66T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 4.23T + 97T^{2} \) |
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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
−8.395403617579795163221193081474, −7.63788575339121410580952568263, −6.54009654170156829444215447648, −6.10078808583365131155021634314, −5.14175780867386362310260472692, −4.53599918670733665115620587964, −3.81216107564850600366774460346, −2.49556571949865015539217498496, −1.97205902740565428137459790785, −0.914853306686238579125614463818,
0.914853306686238579125614463818, 1.97205902740565428137459790785, 2.49556571949865015539217498496, 3.81216107564850600366774460346, 4.53599918670733665115620587964, 5.14175780867386362310260472692, 6.10078808583365131155021634314, 6.54009654170156829444215447648, 7.63788575339121410580952568263, 8.395403617579795163221193081474
