| L(s) = 1 | + 2-s + 4-s − 0.572·5-s + 2.77·7-s + 8-s − 0.572·10-s + 4.01·11-s + 6.90·13-s + 2.77·14-s + 16-s + 4.35·17-s − 6.24·19-s − 0.572·20-s + 4.01·22-s + 23-s − 4.67·25-s + 6.90·26-s + 2.77·28-s − 1.01·29-s + 1.28·31-s + 32-s + 4.35·34-s − 1.59·35-s − 4.23·37-s − 6.24·38-s − 0.572·40-s + 2.98·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.256·5-s + 1.05·7-s + 0.353·8-s − 0.181·10-s + 1.21·11-s + 1.91·13-s + 0.742·14-s + 0.250·16-s + 1.05·17-s − 1.43·19-s − 0.128·20-s + 0.856·22-s + 0.208·23-s − 0.934·25-s + 1.35·26-s + 0.525·28-s − 0.189·29-s + 0.231·31-s + 0.176·32-s + 0.746·34-s − 0.269·35-s − 0.696·37-s − 1.01·38-s − 0.0905·40-s + 0.465·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.955293956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.955293956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 0.572T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 - 4.01T + 11T^{2} \) |
| 13 | \( 1 - 6.90T + 13T^{2} \) |
| 17 | \( 1 - 4.35T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 29 | \( 1 + 1.01T + 29T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 - 2.98T + 41T^{2} \) |
| 43 | \( 1 + 3.64T + 43T^{2} \) |
| 47 | \( 1 - 7.04T + 47T^{2} \) |
| 53 | \( 1 - 6.26T + 53T^{2} \) |
| 59 | \( 1 - 9.57T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 4.51T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 7.93T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 4.22T + 89T^{2} \) |
| 97 | \( 1 - 2.63T + 97T^{2} \) |
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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
−8.539959609236855377749700145510, −7.77408001290468026295964578404, −6.94470075946848708760379303960, −6.07264794416134126764979883576, −5.67933438171575636314884580843, −4.47218342874310303244543080320, −4.01176717588496263195704345918, −3.28745035450132358315132612591, −1.87955187001762128236628182582, −1.19310435517364644008294208724,
1.19310435517364644008294208724, 1.87955187001762128236628182582, 3.28745035450132358315132612591, 4.01176717588496263195704345918, 4.47218342874310303244543080320, 5.67933438171575636314884580843, 6.07264794416134126764979883576, 6.94470075946848708760379303960, 7.77408001290468026295964578404, 8.539959609236855377749700145510
