L(s) = 1 | − 2·4-s + 4.10·7-s − 2.92·13-s + 4·16-s − 8.70·19-s − 5·25-s − 8.21·28-s − 7.31·31-s + 3.20·37-s + 11.6·43-s + 9.85·49-s + 5.84·52-s + 15.5·61-s − 8·64-s − 14.4·67-s − 17·73-s + 17.4·76-s − 17·79-s − 11.9·91-s − 19.6·97-s + 10·100-s + 0.0614·103-s − 17·109-s + 16.4·112-s + ⋯ |
L(s) = 1 | − 4-s + 1.55·7-s − 0.810·13-s + 16-s − 1.99·19-s − 25-s − 1.55·28-s − 1.31·31-s + 0.527·37-s + 1.77·43-s + 1.40·49-s + 0.810·52-s + 1.98·61-s − 64-s − 1.76·67-s − 1.98·73-s + 1.99·76-s − 1.91·79-s − 1.25·91-s − 1.99·97-s + 100-s + 0.00605·103-s − 1.62·109-s + 1.55·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 4.10T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2.92T + 13T^{2} \) |
| 19 | \( 1 + 8.70T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 17T + 73T^{2} \) |
| 79 | \( 1 + 17T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 19.6T + 97T^{2} \) |
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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
−8.529575515352050962439380990363, −7.86037818461606412232832243205, −7.22357449085636630115895994832, −5.91723038894571933480896323168, −5.30437073221507047988847198823, −4.33100049223127836703329506345, −4.12686674135637198316786358476, −2.49831932494473652100922788891, −1.53672575810975349432797573034, 0,
1.53672575810975349432797573034, 2.49831932494473652100922788891, 4.12686674135637198316786358476, 4.33100049223127836703329506345, 5.30437073221507047988847198823, 5.91723038894571933480896323168, 7.22357449085636630115895994832, 7.86037818461606412232832243205, 8.529575515352050962439380990363