| L(s) = 1 | + 4.39·2-s + 3·3-s + 11.2·4-s − 5·5-s + 13.1·6-s + 31.5·7-s + 14.3·8-s + 9·9-s − 21.9·10-s − 5.28·11-s + 33.8·12-s − 5.98·13-s + 138.·14-s − 15·15-s − 27.0·16-s + 84.6·17-s + 39.5·18-s + 57.3·19-s − 56.3·20-s + 94.6·21-s − 23.2·22-s + 31.0·23-s + 43.1·24-s + 25·25-s − 26.2·26-s + 27·27-s + 355.·28-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 0.577·3-s + 1.40·4-s − 0.447·5-s + 0.896·6-s + 1.70·7-s + 0.636·8-s + 0.333·9-s − 0.694·10-s − 0.144·11-s + 0.813·12-s − 0.127·13-s + 2.64·14-s − 0.258·15-s − 0.422·16-s + 1.20·17-s + 0.517·18-s + 0.692·19-s − 0.630·20-s + 0.983·21-s − 0.225·22-s + 0.281·23-s + 0.367·24-s + 0.200·25-s − 0.198·26-s + 0.192·27-s + 2.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.170905597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.170905597\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 4.39T + 8T^{2} \) |
| 7 | \( 1 - 31.5T + 343T^{2} \) |
| 11 | \( 1 + 5.28T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.98T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 57.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 31.0T + 1.21e4T^{2} \) |
| 31 | \( 1 + 14.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 7.76T + 5.06e4T^{2} \) |
| 41 | \( 1 + 399.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 17.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 64.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 122.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 264.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 622.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 327.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 833.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 666.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 189.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 137.T + 9.12e5T^{2} \) |
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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
−11.20779340759601399211642415999, −10.01085587433024632325871053068, −8.610778346901755367248132164311, −7.84230926907683717380523110891, −6.98366238561003080368965341153, −5.47592683839105187865401654376, −4.88801250286545525653114308774, −3.89758150034432843763641614958, −2.88660655055716560795087740868, −1.52627884677992010900054700058,
1.52627884677992010900054700058, 2.88660655055716560795087740868, 3.89758150034432843763641614958, 4.88801250286545525653114308774, 5.47592683839105187865401654376, 6.98366238561003080368965341153, 7.84230926907683717380523110891, 8.610778346901755367248132164311, 10.01085587433024632325871053068, 11.20779340759601399211642415999
