| L(s) = 1 | − 3-s − 4·7-s + 9-s + 13-s + 5·19-s + 4·21-s − 27-s − 3·29-s + 4·31-s − 7·37-s − 39-s + 3·41-s − 2·43-s + 9·47-s + 9·49-s + 9·53-s − 5·57-s + 6·59-s − 8·61-s − 4·63-s − 5·67-s + 3·71-s + 4·73-s − 11·79-s + 81-s + 6·83-s + 3·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s + 1.14·19-s + 0.872·21-s − 0.192·27-s − 0.557·29-s + 0.718·31-s − 1.15·37-s − 0.160·39-s + 0.468·41-s − 0.304·43-s + 1.31·47-s + 9/7·49-s + 1.23·53-s − 0.662·57-s + 0.781·59-s − 1.02·61-s − 0.503·63-s − 0.610·67-s + 0.356·71-s + 0.468·73-s − 1.23·79-s + 1/9·81-s + 0.658·83-s + 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.53815297796095, −13.71149990272443, −13.56173959763670, −13.04655704121643, −12.40168871746775, −12.01069571869315, −11.69455327763257, −10.76356093275535, −10.56540761696491, −9.926428090180491, −9.390202755376207, −9.136000784602020, −8.323989781984204, −7.697487016509011, −6.982546871697780, −6.779966060748384, −6.097878297241702, −5.516091749809496, −5.227039927001400, −4.142361634787491, −3.852927181963050, −3.040119490537450, −2.624337040002026, −1.557048371891352, −0.7951183347238858, 0,
0.7951183347238858, 1.557048371891352, 2.624337040002026, 3.040119490537450, 3.852927181963050, 4.142361634787491, 5.227039927001400, 5.516091749809496, 6.097878297241702, 6.779966060748384, 6.982546871697780, 7.697487016509011, 8.323989781984204, 9.136000784602020, 9.390202755376207, 9.926428090180491, 10.56540761696491, 10.76356093275535, 11.69455327763257, 12.01069571869315, 12.40168871746775, 13.04655704121643, 13.56173959763670, 13.71149990272443, 14.53815297796095
