| L(s) = 1 | − 3-s + 5-s − 4.46·7-s + 9-s − 5.90·11-s − 2.55·13-s − 15-s − 2.72·17-s + 3.14·19-s + 4.46·21-s − 5.52·23-s + 25-s − 27-s − 0.163·29-s − 9.75·31-s + 5.90·33-s − 4.46·35-s − 4.56·37-s + 2.55·39-s + 1.87·41-s − 2.98·43-s + 45-s − 0.397·47-s + 12.8·49-s + 2.72·51-s − 6.93·53-s − 5.90·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.68·7-s + 0.333·9-s − 1.78·11-s − 0.707·13-s − 0.258·15-s − 0.662·17-s + 0.720·19-s + 0.973·21-s − 1.15·23-s + 0.200·25-s − 0.192·27-s − 0.0304·29-s − 1.75·31-s + 1.02·33-s − 0.753·35-s − 0.750·37-s + 0.408·39-s + 0.292·41-s − 0.455·43-s + 0.149·45-s − 0.0580·47-s + 1.84·49-s + 0.382·51-s − 0.952·53-s − 0.796·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1737132093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1737132093\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 4.46T + 7T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 13 | \( 1 + 2.55T + 13T^{2} \) |
| 17 | \( 1 + 2.72T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 + 0.163T + 29T^{2} \) |
| 31 | \( 1 + 9.75T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 + 2.98T + 43T^{2} \) |
| 47 | \( 1 + 0.397T + 47T^{2} \) |
| 53 | \( 1 + 6.93T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 9.41T + 61T^{2} \) |
| 71 | \( 1 - 0.360T + 71T^{2} \) |
| 73 | \( 1 + 9.38T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 7.61T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.52975206626410638410545354296, −7.18384309282388660885226595004, −6.36309683838230626357127930227, −5.69444689744021581810147286562, −5.29586433274662278979172836693, −4.36883297469224267079933676198, −3.36310673288349956443570713154, −2.72365358474142937304929659290, −1.87764061454748796186101457992, −0.19603975036898664007149337332,
0.19603975036898664007149337332, 1.87764061454748796186101457992, 2.72365358474142937304929659290, 3.36310673288349956443570713154, 4.36883297469224267079933676198, 5.29586433274662278979172836693, 5.69444689744021581810147286562, 6.36309683838230626357127930227, 7.18384309282388660885226595004, 7.52975206626410638410545354296
