| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 12-s + 13-s + 14-s + 16-s + 3·17-s + 18-s + 19-s + 21-s + 23-s + 24-s − 5·25-s + 26-s + 27-s + 28-s + 29-s + 5·31-s + 32-s + 3·34-s + 36-s + 10·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.229·19-s + 0.218·21-s + 0.208·23-s + 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.185·29-s + 0.898·31-s + 0.176·32-s + 0.514·34-s + 1/6·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.575739788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.575739788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.58788037796035, −13.33620411079244, −12.59826265096122, −12.24219611112268, −11.73853764384363, −11.28717187996974, −10.69957333508610, −10.24314268050796, −9.692323760337661, −9.188007949933010, −8.646049231684267, −8.002418653934316, −7.604103329547179, −7.300525310182567, −6.444438531395459, −5.971769576756790, −5.568204412752023, −4.837639605071885, −4.287670048836392, −3.899895025035919, −3.230670521010797, −2.599071009098763, −2.198713709814530, −1.281130818950306, −0.7751058253410237,
0.7751058253410237, 1.281130818950306, 2.198713709814530, 2.599071009098763, 3.230670521010797, 3.899895025035919, 4.287670048836392, 4.837639605071885, 5.568204412752023, 5.971769576756790, 6.444438531395459, 7.300525310182567, 7.604103329547179, 8.002418653934316, 8.646049231684267, 9.188007949933010, 9.692323760337661, 10.24314268050796, 10.69957333508610, 11.28717187996974, 11.73853764384363, 12.24219611112268, 12.59826265096122, 13.33620411079244, 13.58788037796035
