| L(s) = 1 | − 3-s + 9-s + 2·13-s − 6·17-s − 4·19-s − 27-s − 6·29-s − 4·31-s − 2·37-s − 2·39-s − 6·41-s + 8·43-s + 12·47-s + 6·51-s − 6·53-s + 4·57-s − 12·59-s − 2·61-s + 8·67-s + 14·73-s + 16·79-s + 81-s − 12·83-s + 6·87-s − 6·89-s + 4·93-s + 14·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.554·13-s − 1.45·17-s − 0.917·19-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 0.840·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s − 0.256·61-s + 0.977·67-s + 1.63·73-s + 1.80·79-s + 1/9·81-s − 1.31·83-s + 0.643·87-s − 0.635·89-s + 0.414·93-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.57305299455631, −14.03632009634668, −13.49441956184317, −13.05506770183275, −12.51240719897553, −12.19229254474687, −11.31150478565147, −11.01295982986198, −10.75744094286019, −10.08630915714146, −9.320745011279580, −8.988284410410529, −8.486376795659378, −7.716670893632567, −7.256277457027396, −6.574264432300895, −6.210377946141902, −5.629803981409619, −4.978407250038501, −4.367159045726463, −3.884974472073815, −3.214967721953716, −2.156454216016891, −1.893085240563413, −0.7888808143694156, 0,
0.7888808143694156, 1.893085240563413, 2.156454216016891, 3.214967721953716, 3.884974472073815, 4.367159045726463, 4.978407250038501, 5.629803981409619, 6.210377946141902, 6.574264432300895, 7.256277457027396, 7.716670893632567, 8.486376795659378, 8.988284410410529, 9.320745011279580, 10.08630915714146, 10.75744094286019, 11.01295982986198, 11.31150478565147, 12.19229254474687, 12.51240719897553, 13.05506770183275, 13.49441956184317, 14.03632009634668, 14.57305299455631
