| L(s) = 1 | + 2.58·2-s − 3·3-s − 1.32·4-s + 5·5-s − 7.75·6-s − 26.6·7-s − 24.0·8-s + 9·9-s + 12.9·10-s − 8.03·11-s + 3.97·12-s + 49.9·13-s − 68.8·14-s − 15·15-s − 51.6·16-s + 116.·17-s + 23.2·18-s + 22.6·19-s − 6.62·20-s + 79.9·21-s − 20.7·22-s + 38.2·23-s + 72.2·24-s + 25·25-s + 129.·26-s − 27·27-s + 35.2·28-s + ⋯ |
| L(s) = 1 | + 0.913·2-s − 0.577·3-s − 0.165·4-s + 0.447·5-s − 0.527·6-s − 1.43·7-s − 1.06·8-s + 0.333·9-s + 0.408·10-s − 0.220·11-s + 0.0955·12-s + 1.06·13-s − 1.31·14-s − 0.258·15-s − 0.807·16-s + 1.66·17-s + 0.304·18-s + 0.273·19-s − 0.0740·20-s + 0.830·21-s − 0.201·22-s + 0.346·23-s + 0.614·24-s + 0.200·25-s + 0.973·26-s − 0.192·27-s + 0.238·28-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
| \(L(2)\) |
\(\approx\) |
\(1.931477151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.931477151\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.58T + 8T^{2} \) |
| 7 | \( 1 + 26.6T + 343T^{2} \) |
| 11 | \( 1 + 8.03T + 1.33e3T^{2} \) |
| 13 | \( 1 - 49.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 22.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.2T + 1.21e4T^{2} \) |
| 31 | \( 1 + 8.82T + 2.97e4T^{2} \) |
| 37 | \( 1 - 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 305.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 275.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 479.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 496.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 563.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 217.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 628.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 682.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 668.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 798.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 867.T + 9.12e5T^{2} \) |
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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
−10.79095301628888046066553244192, −9.744160915013991632398296309496, −9.266528422086008508313615462112, −7.84375906270794337778659211244, −6.38547208765394690340334168138, −6.00434091800514448905695958668, −5.06425670588214396656781129157, −3.75858753725271660509512778420, −2.97281543606151884408825002353, −0.808166535542324138544707266570,
0.808166535542324138544707266570, 2.97281543606151884408825002353, 3.75858753725271660509512778420, 5.06425670588214396656781129157, 6.00434091800514448905695958668, 6.38547208765394690340334168138, 7.84375906270794337778659211244, 9.266528422086008508313615462112, 9.744160915013991632398296309496, 10.79095301628888046066553244192
