L(s) = 1 | − 2.55·2-s − 3-s + 4.50·4-s + 5-s + 2.55·6-s − 0.898·7-s − 6.39·8-s + 9-s − 2.55·10-s − 0.625·11-s − 4.50·12-s − 2.29·13-s + 2.29·14-s − 15-s + 7.28·16-s − 4.00·17-s − 2.55·18-s + 1.74·19-s + 4.50·20-s + 0.898·21-s + 1.59·22-s + 6.16·23-s + 6.39·24-s + 25-s + 5.85·26-s − 27-s − 4.04·28-s + ⋯ |
L(s) = 1 | − 1.80·2-s − 0.577·3-s + 2.25·4-s + 0.447·5-s + 1.04·6-s − 0.339·7-s − 2.25·8-s + 0.333·9-s − 0.806·10-s − 0.188·11-s − 1.30·12-s − 0.636·13-s + 0.612·14-s − 0.258·15-s + 1.82·16-s − 0.971·17-s − 0.601·18-s + 0.399·19-s + 1.00·20-s + 0.196·21-s + 0.340·22-s + 1.28·23-s + 1.30·24-s + 0.200·25-s + 1.14·26-s − 0.192·27-s − 0.765·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4827686771\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4827686771\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 7 | \( 1 + 0.898T + 7T^{2} \) |
| 11 | \( 1 + 0.625T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 + 4.00T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 - 7.13T + 29T^{2} \) |
| 31 | \( 1 + 4.41T + 31T^{2} \) |
| 37 | \( 1 + 7.92T + 37T^{2} \) |
| 41 | \( 1 + 0.859T + 41T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 - 3.14T + 47T^{2} \) |
| 53 | \( 1 - 1.80T + 53T^{2} \) |
| 59 | \( 1 + 2.30T + 59T^{2} \) |
| 61 | \( 1 - 8.68T + 61T^{2} \) |
| 67 | \( 1 - 4.71T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 + 2.79T + 83T^{2} \) |
| 89 | \( 1 - 9.94T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−8.244090159423037456345209374285, −7.28344388360296580054798303714, −6.91696074663958647411602208994, −6.33164581794892479537292199815, −5.40614892605251435330433507786, −4.65933890425035597303991270692, −3.23210201664901949622457801789, −2.40842549826277929491334220351, −1.53432420873091787466662459641, −0.50317019208692402422846756675,
0.50317019208692402422846756675, 1.53432420873091787466662459641, 2.40842549826277929491334220351, 3.23210201664901949622457801789, 4.65933890425035597303991270692, 5.40614892605251435330433507786, 6.33164581794892479537292199815, 6.91696074663958647411602208994, 7.28344388360296580054798303714, 8.244090159423037456345209374285