L(s) = 1 | + 3-s − 5-s + 1.52·7-s + 9-s − 3.59·11-s + 5.59·13-s − 15-s + 4.07·17-s − 1.59·19-s + 1.52·21-s + 23-s + 25-s + 27-s − 4.07·29-s + 2.47·31-s − 3.59·33-s − 1.52·35-s + 9.66·37-s + 5.59·39-s + 5.01·41-s + 7.05·43-s − 45-s − 4.54·47-s − 4.66·49-s + 4.07·51-s + 5.01·53-s + 3.59·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.576·7-s + 0.333·9-s − 1.08·11-s + 1.55·13-s − 0.258·15-s + 0.987·17-s − 0.366·19-s + 0.333·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 0.756·29-s + 0.444·31-s − 0.626·33-s − 0.258·35-s + 1.58·37-s + 0.896·39-s + 0.783·41-s + 1.07·43-s − 0.149·45-s − 0.663·47-s − 0.667·49-s + 0.570·51-s + 0.689·53-s + 0.485·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.111264035\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.111264035\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 - 4.07T + 17T^{2} \) |
| 19 | \( 1 + 1.59T + 19T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 9.66T + 37T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 - 7.05T + 43T^{2} \) |
| 47 | \( 1 + 4.54T + 47T^{2} \) |
| 53 | \( 1 - 5.01T + 53T^{2} \) |
| 59 | \( 1 - 4.07T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 + 5.01T + 71T^{2} \) |
| 73 | \( 1 + 8.79T + 73T^{2} \) |
| 79 | \( 1 - 2.94T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−9.495514595785548947709803492480, −8.586356637424984742981632266419, −7.989372098118033589802635454201, −7.48946655346227699858853883078, −6.25210672291659146773954580449, −5.39746182357178917946156614273, −4.33704916549521674567642155819, −3.49862922096268542721387489359, −2.47009396817158378024093374842, −1.10076524099316238344841236551,
1.10076524099316238344841236551, 2.47009396817158378024093374842, 3.49862922096268542721387489359, 4.33704916549521674567642155819, 5.39746182357178917946156614273, 6.25210672291659146773954580449, 7.48946655346227699858853883078, 7.989372098118033589802635454201, 8.586356637424984742981632266419, 9.495514595785548947709803492480