L(s) = 1 | − 3-s − 2·4-s + 5-s + 7-s − 2·9-s − 3·11-s + 2·12-s + 5·13-s − 15-s + 4·16-s + 3·17-s − 2·19-s − 2·20-s − 21-s + 6·23-s + 25-s + 5·27-s − 2·28-s + 3·29-s + 4·31-s + 3·33-s + 35-s + 4·36-s + 2·37-s − 5·39-s + 12·41-s − 10·43-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s + 1.38·13-s − 0.258·15-s + 16-s + 0.727·17-s − 0.458·19-s − 0.447·20-s − 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.377·28-s + 0.557·29-s + 0.718·31-s + 0.522·33-s + 0.169·35-s + 2/3·36-s + 0.328·37-s − 0.800·39-s + 1.87·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 53 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.85346950558918, −13.67570533130117, −12.95983655125841, −12.71537416840034, −12.13288847272401, −11.48463019222317, −10.95268501360649, −10.67994076881908, −10.13667863992613, −9.586195017775741, −8.939619175086594, −8.547211800953863, −8.207760904469649, −7.613517284064230, −6.881631340689051, −6.151652884239162, −5.767942911398682, −5.423030522016237, −4.751145341690580, −4.407973423891949, −3.553564399221230, −2.987518718978241, −2.421565215462578, −1.226726781240785, −0.9569845106214991, 0,
0.9569845106214991, 1.226726781240785, 2.421565215462578, 2.987518718978241, 3.553564399221230, 4.407973423891949, 4.751145341690580, 5.423030522016237, 5.767942911398682, 6.151652884239162, 6.881631340689051, 7.613517284064230, 8.207760904469649, 8.547211800953863, 8.939619175086594, 9.586195017775741, 10.13667863992613, 10.67994076881908, 10.95268501360649, 11.48463019222317, 12.13288847272401, 12.71537416840034, 12.95983655125841, 13.67570533130117, 13.85346950558918