| L(s) = 1 | − 3.84·2-s + 6.79·4-s − 5·5-s + 4.64·8-s + 19.2·10-s − 23.1·11-s − 61.0·13-s − 72.1·16-s + 0.688·17-s − 63.2·19-s − 33.9·20-s + 88.8·22-s − 124.·23-s + 25·25-s + 234.·26-s − 104.·29-s − 280.·31-s + 240.·32-s − 2.64·34-s − 263.·37-s + 243.·38-s − 23.2·40-s − 243.·41-s + 172.·43-s − 157.·44-s + 478.·46-s − 107.·47-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.849·4-s − 0.447·5-s + 0.205·8-s + 0.608·10-s − 0.633·11-s − 1.30·13-s − 1.12·16-s + 0.00981·17-s − 0.763·19-s − 0.379·20-s + 0.861·22-s − 1.12·23-s + 0.200·25-s + 1.77·26-s − 0.667·29-s − 1.62·31-s + 1.32·32-s − 0.0133·34-s − 1.17·37-s + 1.03·38-s − 0.0917·40-s − 0.927·41-s + 0.611·43-s − 0.537·44-s + 1.53·46-s − 0.333·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01847656318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01847656318\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.84T + 8T^{2} \) |
| 11 | \( 1 + 23.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 0.688T + 4.91e3T^{2} \) |
| 19 | \( 1 + 63.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 104.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 280.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 263.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 243.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 172.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 107.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 44.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 457.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 473.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 229.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 407.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 348.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 840.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 885.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 856.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 189.T + 9.12e5T^{2} \) |
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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
−8.699464698540310121878345359996, −7.977606760020785104741097135406, −7.42348642426776714648589810319, −6.80158759896762515062025701958, −5.57106822568007402636032468987, −4.71088048107707895492070923874, −3.74893364454031843804582839663, −2.44066728638537630956317618515, −1.67062816544532574225409366010, −0.07323650385061304384552562244,
0.07323650385061304384552562244, 1.67062816544532574225409366010, 2.44066728638537630956317618515, 3.74893364454031843804582839663, 4.71088048107707895492070923874, 5.57106822568007402636032468987, 6.80158759896762515062025701958, 7.42348642426776714648589810319, 7.977606760020785104741097135406, 8.699464698540310121878345359996
