| L(s) = 1 | − 1.81·2-s + 3.10·3-s + 1.28·4-s − 2.81·5-s − 5.62·6-s + 1.28·8-s + 6.62·9-s + 5.10·10-s + 3.10·11-s + 3.99·12-s − 13-s − 8.72·15-s − 4.91·16-s + 0.524·17-s − 12.0·18-s − 0.813·19-s − 3.62·20-s − 5.62·22-s + 7.33·23-s + 4.00·24-s + 2.91·25-s + 1.81·26-s + 11.2·27-s + 8.28·29-s + 15.8·30-s − 1.39·31-s + 6.33·32-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 1.79·3-s + 0.644·4-s − 1.25·5-s − 2.29·6-s + 0.455·8-s + 2.20·9-s + 1.61·10-s + 0.935·11-s + 1.15·12-s − 0.277·13-s − 2.25·15-s − 1.22·16-s + 0.127·17-s − 2.83·18-s − 0.186·19-s − 0.811·20-s − 1.19·22-s + 1.53·23-s + 0.816·24-s + 0.583·25-s + 0.355·26-s + 2.16·27-s + 1.53·29-s + 2.89·30-s − 0.250·31-s + 1.12·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.226309462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.226309462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 17 | \( 1 - 0.524T + 17T^{2} \) |
| 19 | \( 1 + 0.813T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 31 | \( 1 + 1.39T + 31T^{2} \) |
| 37 | \( 1 + 6.15T + 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 - 6.75T + 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 + 2.49T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 8.72T + 71T^{2} \) |
| 73 | \( 1 - 2.34T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 1.18T + 97T^{2} \) |
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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
−10.22777885724091869896356456196, −9.341630560827381252799713591963, −8.751367970145830128439756053077, −8.262540967048320716968127345186, −7.38550888749104813822095304232, −6.94361741193310573900581815702, −4.55818398130437604350524277279, −3.77491097616630385233766186485, −2.64983576598868699160395521306, −1.16643628535130692932801266828,
1.16643628535130692932801266828, 2.64983576598868699160395521306, 3.77491097616630385233766186485, 4.55818398130437604350524277279, 6.94361741193310573900581815702, 7.38550888749104813822095304232, 8.262540967048320716968127345186, 8.751367970145830128439756053077, 9.341630560827381252799713591963, 10.22777885724091869896356456196
