L(s) = 1 | − 3-s + 7-s + 9-s − 13-s + 19-s − 21-s − 27-s + 31-s + 2·37-s + 39-s + 43-s − 57-s − 61-s + 63-s + 67-s + 2·73-s − 2·79-s + 81-s − 91-s − 93-s − 97-s − 2·103-s − 109-s − 2·111-s − 117-s + ⋯ |
L(s) = 1 | − 3-s + 7-s + 9-s − 13-s + 19-s − 21-s − 27-s + 31-s + 2·37-s + 39-s + 43-s − 57-s − 61-s + 63-s + 67-s + 2·73-s − 2·79-s + 81-s − 91-s − 93-s − 97-s − 2·103-s − 109-s − 2·111-s − 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8670312834\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8670312834\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.950126173511207971518112912092, −9.391846209416087629579946162636, −8.052275601766956634635657358492, −7.52314732748779362199215745432, −6.58747390409545588061962370111, −5.60654943959490241212002341952, −4.90021730363715398245167051424, −4.17904794327763203216460541414, −2.57929737615652012801569709618, −1.18698551234494244795327854849,
1.18698551234494244795327854849, 2.57929737615652012801569709618, 4.17904794327763203216460541414, 4.90021730363715398245167051424, 5.60654943959490241212002341952, 6.58747390409545588061962370111, 7.52314732748779362199215745432, 8.052275601766956634635657358492, 9.391846209416087629579946162636, 9.950126173511207971518112912092