| L(s) = 1 | − 0.504·2-s + 4.26·3-s − 7.74·4-s − 2.15·6-s + 7·7-s + 7.94·8-s − 8.82·9-s + 54.8·11-s − 33.0·12-s + 16.0·13-s − 3.53·14-s + 57.9·16-s − 0.422·17-s + 4.45·18-s + 127.·19-s + 29.8·21-s − 27.7·22-s + 51.1·23-s + 33.8·24-s − 8.08·26-s − 152.·27-s − 54.2·28-s + 41.4·29-s + 192.·31-s − 92.8·32-s + 233.·33-s + 0.213·34-s + ⋯ |
| L(s) = 1 | − 0.178·2-s + 0.820·3-s − 0.968·4-s − 0.146·6-s + 0.377·7-s + 0.351·8-s − 0.326·9-s + 1.50·11-s − 0.794·12-s + 0.341·13-s − 0.0674·14-s + 0.905·16-s − 0.00602·17-s + 0.0583·18-s + 1.53·19-s + 0.310·21-s − 0.268·22-s + 0.463·23-s + 0.288·24-s − 0.0609·26-s − 1.08·27-s − 0.365·28-s + 0.265·29-s + 1.11·31-s − 0.512·32-s + 1.23·33-s + 0.00107·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.830929131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.830929131\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 0.504T + 8T^{2} \) |
| 3 | \( 1 - 4.26T + 27T^{2} \) |
| 11 | \( 1 - 54.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 0.422T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 51.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 192.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 76.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 294.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 540.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 661.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 410.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 46.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 10.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 491.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 814.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 858.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 341.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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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
−12.26053526007075512507719814418, −11.33028578209523573557695785688, −9.892273726653195028222975289733, −9.049780431728389803409956612483, −8.473298194236115450889964792591, −7.32225590059572440674240905108, −5.72724151351234671324246482849, −4.34301677681016364597321912056, −3.21776217946735066482870990446, −1.18988022864534923705269285696,
1.18988022864534923705269285696, 3.21776217946735066482870990446, 4.34301677681016364597321912056, 5.72724151351234671324246482849, 7.32225590059572440674240905108, 8.473298194236115450889964792591, 9.049780431728389803409956612483, 9.892273726653195028222975289733, 11.33028578209523573557695785688, 12.26053526007075512507719814418
