| L(s) = 1 | − 2.64·2-s + 6.66·3-s − 25.0·4-s − 25·5-s − 17.6·6-s + 12.8·7-s + 150.·8-s − 198.·9-s + 66.0·10-s − 166.·12-s + 485.·13-s − 34.0·14-s − 166.·15-s + 402.·16-s − 266.·17-s + 524.·18-s + 149.·19-s + 625.·20-s + 85.8·21-s − 3.21e3·23-s + 1.00e3·24-s + 625·25-s − 1.28e3·26-s − 2.94e3·27-s − 322.·28-s − 2.94e3·29-s + 440.·30-s + ⋯ |
| L(s) = 1 | − 0.467·2-s + 0.427·3-s − 0.781·4-s − 0.447·5-s − 0.199·6-s + 0.0993·7-s + 0.832·8-s − 0.817·9-s + 0.208·10-s − 0.334·12-s + 0.796·13-s − 0.0464·14-s − 0.191·15-s + 0.393·16-s − 0.223·17-s + 0.381·18-s + 0.0951·19-s + 0.349·20-s + 0.0424·21-s − 1.26·23-s + 0.355·24-s + 0.200·25-s − 0.371·26-s − 0.776·27-s − 0.0776·28-s − 0.651·29-s + 0.0893·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8948018315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8948018315\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.64T + 32T^{2} \) |
| 3 | \( 1 - 6.66T + 243T^{2} \) |
| 7 | \( 1 - 12.8T + 1.68e4T^{2} \) |
| 13 | \( 1 - 485.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 266.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 149.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.21e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 808.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.01e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.97e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.12e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.50e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.59e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.27e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.32e5T + 8.58e9T^{2} \) |
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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
−9.687938288064523197167890096646, −8.864191099771209038191500749953, −8.264812775868235978315032746430, −7.63872478912060923491486570895, −6.27889669609778865954084275779, −5.23411303961112226660731322438, −4.11749504103314156394810269505, −3.30816119961775258030121866391, −1.83845906849806565650435669895, −0.48157705469417235621744734632,
0.48157705469417235621744734632, 1.83845906849806565650435669895, 3.30816119961775258030121866391, 4.11749504103314156394810269505, 5.23411303961112226660731322438, 6.27889669609778865954084275779, 7.63872478912060923491486570895, 8.264812775868235978315032746430, 8.864191099771209038191500749953, 9.687938288064523197167890096646
