L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s + 11-s − 2·13-s + 15-s − 19-s − 3·21-s + 6·23-s + 25-s − 27-s − 7·29-s − 10·31-s − 33-s − 3·35-s + 3·37-s + 2·39-s + 9·41-s − 8·43-s − 45-s − 3·47-s + 2·49-s + 5·53-s − 55-s + 57-s − 8·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.258·15-s − 0.229·19-s − 0.654·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.29·29-s − 1.79·31-s − 0.174·33-s − 0.507·35-s + 0.493·37-s + 0.320·39-s + 1.40·41-s − 1.21·43-s − 0.149·45-s − 0.437·47-s + 2/7·49-s + 0.686·53-s − 0.134·55-s + 0.132·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.335177032\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.335177032\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.32620062860698, −13.65176693404308, −12.89881496049104, −12.74426145509787, −12.05240157406638, −11.54539798361632, −11.10767736949867, −10.88964686854111, −10.28029977420673, −9.357663430020841, −9.226264056508603, −8.516043013747892, −7.782826526183952, −7.507690098791554, −7.015263811767067, −6.321819786245644, −5.651871876526275, −5.139851684427895, −4.699124783316511, −4.115068164992989, −3.499372757515517, −2.712976545130155, −1.829247495263167, −1.389303779042060, −0.4068233776536844,
0.4068233776536844, 1.389303779042060, 1.829247495263167, 2.712976545130155, 3.499372757515517, 4.115068164992989, 4.699124783316511, 5.139851684427895, 5.651871876526275, 6.321819786245644, 7.015263811767067, 7.507690098791554, 7.782826526183952, 8.516043013747892, 9.226264056508603, 9.357663430020841, 10.28029977420673, 10.88964686854111, 11.10767736949867, 11.54539798361632, 12.05240157406638, 12.74426145509787, 12.89881496049104, 13.65176693404308, 14.32620062860698