L(s) = 1 | − 1.41·2-s + 3-s + 1.00·4-s − 5-s − 1.41·6-s + 9-s + 1.41·10-s + 1.00·12-s − 15-s − 0.999·16-s − 1.41·18-s + 1.41·19-s − 1.00·20-s + 1.41·23-s + 25-s + 27-s + 1.41·30-s − 1.41·31-s + 1.41·32-s + 1.00·36-s − 2.00·38-s − 45-s − 2.00·46-s − 0.999·48-s − 1.41·50-s + 1.41·53-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3-s + 1.00·4-s − 5-s − 1.41·6-s + 9-s + 1.41·10-s + 1.00·12-s − 15-s − 0.999·16-s − 1.41·18-s + 1.41·19-s − 1.00·20-s + 1.41·23-s + 25-s + 27-s + 1.41·30-s − 1.41·31-s + 1.41·32-s + 1.00·36-s − 2.00·38-s − 45-s − 2.00·46-s − 0.999·48-s − 1.41·50-s + 1.41·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6025652098\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6025652098\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.40773855280116794576302295303, −9.363250945143462697860701583681, −8.978941710016274445341356389273, −8.097276034469993834674201595001, −7.43231271360043508287367320463, −6.94527195981563389294367829869, −5.05119757585646418650918676465, −3.86403498685224914303091667555, −2.81268713147475234175828835137, −1.26452296747627704830958768689,
1.26452296747627704830958768689, 2.81268713147475234175828835137, 3.86403498685224914303091667555, 5.05119757585646418650918676465, 6.94527195981563389294367829869, 7.43231271360043508287367320463, 8.097276034469993834674201595001, 8.978941710016274445341356389273, 9.363250945143462697860701583681, 10.40773855280116794576302295303