| L(s) = 1 | + 3.40·3-s + 5-s + 0.772·7-s + 8.57·9-s − 1.54·11-s + 3.85·13-s + 3.40·15-s − 1.22·17-s − 6.80·19-s + 2.62·21-s − 3.40·23-s + 25-s + 18.9·27-s + 29-s + 9.12·31-s − 5.25·33-s + 0.772·35-s − 0.454·37-s + 13.1·39-s + 3.54·41-s − 3.86·43-s + 8.57·45-s + 8·47-s − 6.40·49-s − 4.17·51-s − 0.318·53-s − 1.54·55-s + ⋯ |
| L(s) = 1 | + 1.96·3-s + 0.447·5-s + 0.292·7-s + 2.85·9-s − 0.466·11-s + 1.06·13-s + 0.878·15-s − 0.297·17-s − 1.56·19-s + 0.573·21-s − 0.709·23-s + 0.200·25-s + 3.65·27-s + 0.185·29-s + 1.63·31-s − 0.915·33-s + 0.130·35-s − 0.0746·37-s + 2.10·39-s + 0.553·41-s − 0.589·43-s + 1.27·45-s + 1.16·47-s − 0.914·49-s − 0.584·51-s − 0.0437·53-s − 0.208·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.264406312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.264406312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.40T + 3T^{2} \) |
| 7 | \( 1 - 0.772T + 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 + 3.40T + 23T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 0.454T + 37T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 + 3.86T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 0.318T + 53T^{2} \) |
| 59 | \( 1 + 8.66T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 9.92T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 1.71T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−8.820454750735147143514062429404, −8.310211366195458854744108496307, −7.83368781583224225048444032266, −6.78515219454673908152273452064, −6.08374924331227186887950810740, −4.62209868021235168920958392737, −4.06987317398517438753492605415, −3.03574733236698846877669742580, −2.29161159261024942255706151586, −1.44370889314726540718880788983,
1.44370889314726540718880788983, 2.29161159261024942255706151586, 3.03574733236698846877669742580, 4.06987317398517438753492605415, 4.62209868021235168920958392737, 6.08374924331227186887950810740, 6.78515219454673908152273452064, 7.83368781583224225048444032266, 8.310211366195458854744108496307, 8.820454750735147143514062429404
