| L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 3·9-s − 2·10-s + 11-s + 14-s + 16-s − 4·17-s − 3·18-s − 2·20-s + 22-s + 4·23-s − 25-s + 28-s + 4·29-s − 4·31-s + 32-s − 4·34-s − 2·35-s − 3·36-s − 4·37-s − 2·40-s − 10·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 9-s − 0.632·10-s + 0.301·11-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.707·18-s − 0.447·20-s + 0.213·22-s + 0.834·23-s − 1/5·25-s + 0.188·28-s + 0.742·29-s − 0.718·31-s + 0.176·32-s − 0.685·34-s − 0.338·35-s − 1/2·36-s − 0.657·37-s − 0.316·40-s − 1.56·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.040955390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.040955390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.22266189672605, −14.87011495876031, −14.20668930191784, −13.80117150858267, −13.25010156960843, −12.56088413399459, −12.05366395796223, −11.54426122692889, −11.18728427828812, −10.74419286407229, −9.968899326500411, −9.130499394107605, −8.587925479278902, −8.176059831908646, −7.485652330505229, −6.809426655731001, −6.422915766277318, −5.511429933442767, −5.062712452978704, −4.418461631959807, −3.734785469630044, −3.217339059530570, −2.445365345226912, −1.656463057163309, −0.4865701807986871,
0.4865701807986871, 1.656463057163309, 2.445365345226912, 3.217339059530570, 3.734785469630044, 4.418461631959807, 5.062712452978704, 5.511429933442767, 6.422915766277318, 6.809426655731001, 7.485652330505229, 8.176059831908646, 8.587925479278902, 9.130499394107605, 9.968899326500411, 10.74419286407229, 11.18728427828812, 11.54426122692889, 12.05366395796223, 12.56088413399459, 13.25010156960843, 13.80117150858267, 14.20668930191784, 14.87011495876031, 15.22266189672605
