| L(s) = 1 | + 3-s + 5-s − 3.58·7-s + 9-s + 0.715·13-s + 15-s − 1.58·17-s − 3.58·21-s + 23-s + 25-s + 27-s + 5.58·29-s + 4.87·31-s − 3.58·35-s + 8.15·37-s + 0.715·39-s + 4.30·41-s + 8.45·43-s + 45-s − 7.17·47-s + 5.87·49-s − 1.58·51-s + 8.30·53-s − 2.30·59-s + 6.45·61-s − 3.58·63-s + 0.715·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.35·7-s + 0.333·9-s + 0.198·13-s + 0.258·15-s − 0.385·17-s − 0.782·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 1.03·29-s + 0.875·31-s − 0.606·35-s + 1.34·37-s + 0.114·39-s + 0.672·41-s + 1.29·43-s + 0.149·45-s − 1.04·47-s + 0.838·49-s − 0.222·51-s + 1.14·53-s − 0.299·59-s + 0.827·61-s − 0.452·63-s + 0.0887·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.124191932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.124191932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 0.715T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 5.58T + 29T^{2} \) |
| 31 | \( 1 - 4.87T + 31T^{2} \) |
| 37 | \( 1 - 8.15T + 37T^{2} \) |
| 41 | \( 1 - 4.30T + 41T^{2} \) |
| 43 | \( 1 - 8.45T + 43T^{2} \) |
| 47 | \( 1 + 7.17T + 47T^{2} \) |
| 53 | \( 1 - 8.30T + 53T^{2} \) |
| 59 | \( 1 + 2.30T + 59T^{2} \) |
| 61 | \( 1 - 6.45T + 61T^{2} \) |
| 67 | \( 1 + 6.15T + 67T^{2} \) |
| 71 | \( 1 + 1.01T + 71T^{2} \) |
| 73 | \( 1 + 5.17T + 73T^{2} \) |
| 79 | \( 1 - 9.89T + 79T^{2} \) |
| 83 | \( 1 - 5.01T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 6.45T + 97T^{2} \) |
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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
−8.965565806736929861061972497438, −8.138658268267212360142584002888, −7.26468172921653688586343477190, −6.43183885832394844288862905047, −6.02403188654542751428581729134, −4.81523533494161858235634675448, −3.93034566409509717222496427382, −2.99314347025409136142181810938, −2.37044060451825392700643278334, −0.883436019144187733132308289500,
0.883436019144187733132308289500, 2.37044060451825392700643278334, 2.99314347025409136142181810938, 3.93034566409509717222496427382, 4.81523533494161858235634675448, 6.02403188654542751428581729134, 6.43183885832394844288862905047, 7.26468172921653688586343477190, 8.138658268267212360142584002888, 8.965565806736929861061972497438
