| L(s) = 1 | + 2.20·2-s − 3.13·4-s + 7·7-s − 24.5·8-s − 56.2·11-s + 38.9·13-s + 15.4·14-s − 29.1·16-s + 119.·17-s − 13.0·19-s − 124.·22-s + 130.·23-s + 85.8·26-s − 21.9·28-s − 77.9·29-s + 61.0·31-s + 132.·32-s + 263.·34-s + 167.·37-s − 28.6·38-s − 436.·41-s − 393.·43-s + 175.·44-s + 288.·46-s − 365.·47-s + 49·49-s − 121.·52-s + ⋯ |
| L(s) = 1 | + 0.780·2-s − 0.391·4-s + 0.377·7-s − 1.08·8-s − 1.54·11-s + 0.829·13-s + 0.294·14-s − 0.455·16-s + 1.70·17-s − 0.157·19-s − 1.20·22-s + 1.18·23-s + 0.647·26-s − 0.147·28-s − 0.499·29-s + 0.353·31-s + 0.730·32-s + 1.32·34-s + 0.743·37-s − 0.122·38-s − 1.66·41-s − 1.39·43-s + 0.602·44-s + 0.923·46-s − 1.13·47-s + 0.142·49-s − 0.324·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 2.20T + 8T^{2} \) |
| 11 | \( 1 + 56.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 13.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 77.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 61.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 282.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 414.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 563.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 103.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 128.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 641.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 512.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 186.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.417645721371742693083756493629, −8.070915316938694127764823271457, −6.98120936355555358687790963403, −5.84415991563978484446451086637, −5.29338877469447337785499449278, −4.63601195322736877787852377934, −3.45622790822870777946258122728, −2.88743705721921583799535882633, −1.33913911791753540280836718145, 0,
1.33913911791753540280836718145, 2.88743705721921583799535882633, 3.45622790822870777946258122728, 4.63601195322736877787852377934, 5.29338877469447337785499449278, 5.84415991563978484446451086637, 6.98120936355555358687790963403, 8.070915316938694127764823271457, 8.417645721371742693083756493629
