| L(s) = 1 | + 2·2-s + 9.38·3-s + 4·4-s − 9.38·5-s + 18.7·6-s + 8·8-s + 61·9-s − 18.7·10-s + 20·11-s + 37.5·12-s − 65.6·13-s − 88·15-s + 16·16-s − 56.2·17-s + 122·18-s − 9.38·19-s − 37.5·20-s + 40·22-s + 48·23-s + 75.0·24-s − 37·25-s − 131.·26-s + 318.·27-s − 166·29-s − 176·30-s + 206.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.80·3-s + 0.5·4-s − 0.839·5-s + 1.27·6-s + 0.353·8-s + 2.25·9-s − 0.593·10-s + 0.548·11-s + 0.902·12-s − 1.40·13-s − 1.51·15-s + 0.250·16-s − 0.803·17-s + 1.59·18-s − 0.113·19-s − 0.419·20-s + 0.387·22-s + 0.435·23-s + 0.638·24-s − 0.295·25-s − 0.990·26-s + 2.27·27-s − 1.06·29-s − 1.07·30-s + 1.19·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.432688519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.432688519\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 9.38T + 27T^{2} \) |
| 5 | \( 1 + 9.38T + 125T^{2} \) |
| 11 | \( 1 - 20T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 56.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 9.38T + 6.85e3T^{2} \) |
| 23 | \( 1 - 48T + 1.21e4T^{2} \) |
| 29 | \( 1 + 166T + 2.43e4T^{2} \) |
| 31 | \( 1 - 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 78T + 5.06e4T^{2} \) |
| 41 | \( 1 + 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 436T + 7.95e4T^{2} \) |
| 47 | \( 1 + 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 62T + 1.48e5T^{2} \) |
| 59 | \( 1 - 666.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 272.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 580T + 3.00e5T^{2} \) |
| 71 | \( 1 + 544T + 3.57e5T^{2} \) |
| 73 | \( 1 - 600.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 680T + 4.93e5T^{2} \) |
| 83 | \( 1 + 196.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 656.T + 9.12e5T^{2} \) |
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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
−13.59963797485483083534632429630, −12.65761964241543160645639262883, −11.60381728625889836671756696456, −10.00587691118305216349327883130, −8.894540399069061869708057750005, −7.79143107836201692503870437619, −6.94647134629169526302865347732, −4.59366733869488539456384880253, −3.56321171856643082977611247433, −2.26949176586125277843156415495,
2.26949176586125277843156415495, 3.56321171856643082977611247433, 4.59366733869488539456384880253, 6.94647134629169526302865347732, 7.79143107836201692503870437619, 8.894540399069061869708057750005, 10.00587691118305216349327883130, 11.60381728625889836671756696456, 12.65761964241543160645639262883, 13.59963797485483083534632429630
