| L(s) = 1 | − 3-s + 4-s − 7-s − 11-s − 12-s + 16-s − 17-s + 21-s + 2·23-s + 25-s + 27-s − 28-s − 29-s − 31-s + 33-s − 37-s + 2·41-s − 44-s − 48-s + 51-s − 59-s − 61-s + 64-s − 68-s − 2·69-s − 75-s + 77-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s − 7-s − 11-s − 12-s + 16-s − 17-s + 21-s + 2·23-s + 25-s + 27-s − 28-s − 29-s − 31-s + 33-s − 37-s + 2·41-s − 44-s − 48-s + 51-s − 59-s − 61-s + 64-s − 68-s − 2·69-s − 75-s + 77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4493265963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4493265963\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−14.86356454031015697333911315034, −13.09589613086253146464545743202, −12.43990908860202224240837164010, −11.01446604629011781551544529494, −10.76555010249983891945786170955, −9.121647327898758088329847107513, −7.30510568839843872365517009292, −6.37890239293184761621864982629, −5.25394517311578314477026475826, −2.91606813106134337451054950253,
2.91606813106134337451054950253, 5.25394517311578314477026475826, 6.37890239293184761621864982629, 7.30510568839843872365517009292, 9.121647327898758088329847107513, 10.76555010249983891945786170955, 11.01446604629011781551544529494, 12.43990908860202224240837164010, 13.09589613086253146464545743202, 14.86356454031015697333911315034
