L(s) = 1 | − 4-s + 7-s + 16-s − 28-s + 2·37-s + 2·43-s + 49-s − 64-s − 2·67-s + 2·79-s − 2·109-s + 112-s + ⋯ |
L(s) = 1 | − 4-s + 7-s + 16-s − 28-s + 2·37-s + 2·43-s + 49-s − 64-s − 2·67-s + 2·79-s − 2·109-s + 112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.001733853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001733853\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.421267289855249510190539319436, −8.920125528939978502443454260671, −7.974778591130946263874335005459, −7.56751889911544202471479912421, −6.21591079942062532022809307022, −5.39374229976948031752166756148, −4.57224141069611571839403928779, −3.93328169845901841106526795227, −2.58292344259865140842019713269, −1.14505708980532822680745106738,
1.14505708980532822680745106738, 2.58292344259865140842019713269, 3.93328169845901841106526795227, 4.57224141069611571839403928779, 5.39374229976948031752166756148, 6.21591079942062532022809307022, 7.56751889911544202471479912421, 7.974778591130946263874335005459, 8.920125528939978502443454260671, 9.421267289855249510190539319436