L(s) = 1 | + 3-s − 2·4-s − 5-s + 7-s + 9-s − 2·12-s − 2·13-s − 15-s + 4·16-s − 6·17-s + 7·19-s + 2·20-s + 21-s − 6·23-s + 25-s + 27-s − 2·28-s − 31-s − 35-s − 2·36-s − 7·37-s − 2·39-s + 6·41-s − 8·43-s − 45-s + 4·48-s − 6·49-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.554·13-s − 0.258·15-s + 16-s − 1.45·17-s + 1.60·19-s + 0.447·20-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.179·31-s − 0.169·35-s − 1/3·36-s − 1.15·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s + 0.577·48-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−8.901173099229814277480267609020, −8.080762717016799727052973771035, −7.60746345376476564860500482297, −6.59523037473790869546358443506, −5.36747921289838822076634820154, −4.65725991292015173841856541050, −3.90838681993568032244631066955, −2.97589293278646371195021990890, −1.63448908436523374581135103796, 0,
1.63448908436523374581135103796, 2.97589293278646371195021990890, 3.90838681993568032244631066955, 4.65725991292015173841856541050, 5.36747921289838822076634820154, 6.59523037473790869546358443506, 7.60746345376476564860500482297, 8.080762717016799727052973771035, 8.901173099229814277480267609020