L(s) = 1 | − 3-s − 2·4-s + 9-s + 5·11-s + 2·12-s − 4·13-s + 4·16-s − 2·17-s − 3·19-s + 23-s − 5·25-s − 27-s − 8·29-s + 10·31-s − 5·33-s − 2·36-s + 4·37-s + 4·39-s − 7·41-s + 10·43-s − 10·44-s + 3·47-s − 4·48-s + 2·51-s + 8·52-s + 3·53-s + 3·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s + 1/3·9-s + 1.50·11-s + 0.577·12-s − 1.10·13-s + 16-s − 0.485·17-s − 0.688·19-s + 0.208·23-s − 25-s − 0.192·27-s − 1.48·29-s + 1.79·31-s − 0.870·33-s − 1/3·36-s + 0.657·37-s + 0.640·39-s − 1.09·41-s + 1.52·43-s − 1.50·44-s + 0.437·47-s − 0.577·48-s + 0.280·51-s + 1.10·52-s + 0.412·53-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.395083056168476340595160662222, −7.42787569264773526747552444546, −6.71712454588677528285323508098, −5.89990296468135663436659382942, −5.18593557076444591765226530347, −4.15421816154601291686783697564, −4.02760721463459717985252559663, −2.48397366998135367015442328975, −1.21676283525296595144680309244, 0,
1.21676283525296595144680309244, 2.48397366998135367015442328975, 4.02760721463459717985252559663, 4.15421816154601291686783697564, 5.18593557076444591765226530347, 5.89990296468135663436659382942, 6.71712454588677528285323508098, 7.42787569264773526747552444546, 8.395083056168476340595160662222