| L(s) = 1 | − 1.46·2-s + 1.61·3-s + 0.151·4-s − 0.466·5-s − 2.37·6-s + 2.71·8-s − 0.381·9-s + 0.684·10-s + 0.244·12-s + 1.58·13-s − 0.755·15-s − 4.27·16-s − 5.22·17-s + 0.560·18-s − 4.22·19-s − 0.0706·20-s − 1.80·23-s + 4.38·24-s − 4.78·25-s − 2.32·26-s − 5.47·27-s + 2.71·29-s + 1.10·30-s − 1.29·31-s + 0.854·32-s + 7.66·34-s − 0.0577·36-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.934·3-s + 0.0756·4-s − 0.208·5-s − 0.968·6-s + 0.958·8-s − 0.127·9-s + 0.216·10-s + 0.0706·12-s + 0.438·13-s − 0.194·15-s − 1.06·16-s − 1.26·17-s + 0.132·18-s − 0.968·19-s − 0.0157·20-s − 0.376·23-s + 0.895·24-s − 0.956·25-s − 0.455·26-s − 1.05·27-s + 0.504·29-s + 0.202·30-s − 0.232·31-s + 0.150·32-s + 1.31·34-s − 0.00963·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9417605328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9417605328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 0.466T + 5T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 + 4.22T + 19T^{2} \) |
| 23 | \( 1 + 1.80T + 23T^{2} \) |
| 29 | \( 1 - 2.71T + 29T^{2} \) |
| 31 | \( 1 + 1.29T + 31T^{2} \) |
| 37 | \( 1 + 1.94T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 - 9.74T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 8.91T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−8.149381627507042159014715065398, −7.84427955831257500471016214243, −6.89834443684568737168227916502, −6.20710296692675601660664317821, −5.15534195303171123962865063557, −4.16138713844793849470745041933, −3.74187746552772742677288891099, −2.43722194024536090519611136699, −1.94238953289155480214602729947, −0.55568503620360162615142324649,
0.55568503620360162615142324649, 1.94238953289155480214602729947, 2.43722194024536090519611136699, 3.74187746552772742677288891099, 4.16138713844793849470745041933, 5.15534195303171123962865063557, 6.20710296692675601660664317821, 6.89834443684568737168227916502, 7.84427955831257500471016214243, 8.149381627507042159014715065398
