L(s) = 1 | + 4.97·2-s + 16.7·4-s − 5.48·7-s + 43.3·8-s − 11·11-s − 24.5·13-s − 27.2·14-s + 81.6·16-s − 59.3·17-s + 5.89·19-s − 54.6·22-s − 68.4·23-s − 122.·26-s − 91.6·28-s + 265.·29-s − 196.·31-s + 59.2·32-s − 295.·34-s − 166.·37-s + 29.2·38-s − 424.·41-s + 177.·43-s − 183.·44-s − 340.·46-s + 141.·47-s − 312.·49-s − 410.·52-s + ⋯ |
L(s) = 1 | + 1.75·2-s + 2.08·4-s − 0.296·7-s + 1.91·8-s − 0.301·11-s − 0.524·13-s − 0.520·14-s + 1.27·16-s − 0.847·17-s + 0.0711·19-s − 0.529·22-s − 0.620·23-s − 0.921·26-s − 0.618·28-s + 1.69·29-s − 1.13·31-s + 0.327·32-s − 1.48·34-s − 0.740·37-s + 0.125·38-s − 1.61·41-s + 0.628·43-s − 0.629·44-s − 1.09·46-s + 0.438·47-s − 0.912·49-s − 1.09·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 - 4.97T + 8T^{2} \) |
| 7 | \( 1 + 5.48T + 343T^{2} \) |
| 13 | \( 1 + 24.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.89T + 6.85e3T^{2} \) |
| 23 | \( 1 + 68.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 265.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 166.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 424.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 177.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 141.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 339.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 662.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 313.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 153.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 153.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 403.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 652.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 959.T + 9.12e5T^{2} \) |
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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
−7.963938549364779013062305766192, −6.99307079737127380877926243053, −6.54626341420216804478150827914, −5.66718950796358975084677542984, −4.95132547434900562174194534244, −4.29426592303396401715113613131, −3.39256703518595098074915982295, −2.63864660540934429319937364955, −1.74607079502606472597538859262, 0,
1.74607079502606472597538859262, 2.63864660540934429319937364955, 3.39256703518595098074915982295, 4.29426592303396401715113613131, 4.95132547434900562174194534244, 5.66718950796358975084677542984, 6.54626341420216804478150827914, 6.99307079737127380877926243053, 7.963938549364779013062305766192