L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s + 12-s + 3·13-s − 16-s − 2·17-s − 18-s + 19-s − 2·23-s − 3·24-s − 3·26-s − 27-s − 8·29-s − 8·31-s − 5·32-s + 2·34-s − 36-s − 7·37-s − 38-s − 3·39-s + 8·43-s + 2·46-s + 10·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.288·12-s + 0.832·13-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.417·23-s − 0.612·24-s − 0.588·26-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.883·32-s + 0.342·34-s − 1/6·36-s − 1.15·37-s − 0.162·38-s − 0.480·39-s + 1.21·43-s + 0.294·46-s + 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−8.352473628106623913027364813151, −7.36094805434549695111842196649, −6.96980614398058522250287141210, −5.65263470222658997680926370661, −5.44683094274408177280819536130, −4.13854069990749647485381286502, −3.79225144094643222973826352921, −2.17796198616348634558110900406, −1.15430294010390086441257495850, 0,
1.15430294010390086441257495850, 2.17796198616348634558110900406, 3.79225144094643222973826352921, 4.13854069990749647485381286502, 5.44683094274408177280819536130, 5.65263470222658997680926370661, 6.96980614398058522250287141210, 7.36094805434549695111842196649, 8.352473628106623913027364813151