| L(s) = 1 | − 2·3-s + 2·7-s + 9-s − 5·13-s − 3·17-s − 2·19-s − 4·21-s + 6·23-s + 4·27-s − 3·29-s − 2·31-s + 7·37-s + 10·39-s + 3·41-s + 8·43-s + 6·47-s − 3·49-s + 6·51-s + 3·53-s + 4·57-s − 10·61-s + 2·63-s − 10·67-s − 12·69-s − 12·71-s − 14·73-s − 2·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s − 1.38·13-s − 0.727·17-s − 0.458·19-s − 0.872·21-s + 1.25·23-s + 0.769·27-s − 0.557·29-s − 0.359·31-s + 1.15·37-s + 1.60·39-s + 0.468·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.840·51-s + 0.412·53-s + 0.529·57-s − 1.28·61-s + 0.251·63-s − 1.22·67-s − 1.44·69-s − 1.42·71-s − 1.63·73-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−14.76748285486329, −14.56128071280550, −13.72923733094078, −13.18987796417211, −12.57775042691699, −12.25307274587526, −11.65451614502509, −11.12033932367571, −10.92168495350740, −10.32231612407444, −9.688517163989766, −8.941070358594587, −8.752908303525933, −7.632398231880693, −7.473442355030950, −6.845083787976486, −6.053557772194286, −5.758236012492613, −4.990440591671363, −4.607734104827978, −4.204471811358460, −3.013690751559558, −2.481951369134435, −1.674502034226112, −0.7899650103012130, 0,
0.7899650103012130, 1.674502034226112, 2.481951369134435, 3.013690751559558, 4.204471811358460, 4.607734104827978, 4.990440591671363, 5.758236012492613, 6.053557772194286, 6.845083787976486, 7.473442355030950, 7.632398231880693, 8.752908303525933, 8.941070358594587, 9.688517163989766, 10.32231612407444, 10.92168495350740, 11.12033932367571, 11.65451614502509, 12.25307274587526, 12.57775042691699, 13.18987796417211, 13.72923733094078, 14.56128071280550, 14.76748285486329
