| L(s) = 1 | + 2.76·3-s + 5-s + 3.75·7-s + 4.65·9-s + 1.54·11-s + 4.15·13-s + 2.76·15-s + 5.74·17-s − 7.03·19-s + 10.3·21-s + 6.77·23-s + 25-s + 4.57·27-s − 10.1·29-s − 0.0578·31-s + 4.26·33-s + 3.75·35-s − 2.18·37-s + 11.4·39-s − 8.82·41-s + 6.08·43-s + 4.65·45-s − 4.08·47-s + 7.07·49-s + 15.8·51-s − 3.49·53-s + 1.54·55-s + ⋯ |
| L(s) = 1 | + 1.59·3-s + 0.447·5-s + 1.41·7-s + 1.55·9-s + 0.464·11-s + 1.15·13-s + 0.714·15-s + 1.39·17-s − 1.61·19-s + 2.26·21-s + 1.41·23-s + 0.200·25-s + 0.880·27-s − 1.88·29-s − 0.0103·31-s + 0.742·33-s + 0.634·35-s − 0.358·37-s + 1.83·39-s − 1.37·41-s + 0.927·43-s + 0.693·45-s − 0.595·47-s + 1.01·49-s + 2.22·51-s − 0.480·53-s + 0.207·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.640650178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.640650178\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 3 | \( 1 - 2.76T + 3T^{2} \) |
| 7 | \( 1 - 3.75T + 7T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 + 7.03T + 19T^{2} \) |
| 23 | \( 1 - 6.77T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 0.0578T + 31T^{2} \) |
| 37 | \( 1 + 2.18T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 6.08T + 43T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 + 3.49T + 53T^{2} \) |
| 59 | \( 1 - 9.20T + 59T^{2} \) |
| 61 | \( 1 - 8.32T + 61T^{2} \) |
| 67 | \( 1 + 3.93T + 67T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 8.83T + 79T^{2} \) |
| 83 | \( 1 - 9.09T + 83T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−8.196548234504391953595264074034, −7.42962389845759923597562134132, −6.75625333190279845978940910893, −5.74101480921452490765134494691, −5.06003462513115601064472215519, −4.05985222503685329873213714285, −3.60714236451057779761903234441, −2.66923942084185713964151799519, −1.73726720874122651705736386436, −1.35057193315269857911918093889,
1.35057193315269857911918093889, 1.73726720874122651705736386436, 2.66923942084185713964151799519, 3.60714236451057779761903234441, 4.05985222503685329873213714285, 5.06003462513115601064472215519, 5.74101480921452490765134494691, 6.75625333190279845978940910893, 7.42962389845759923597562134132, 8.196548234504391953595264074034
