| L(s) = 1 | − 1.61·2-s + 0.618·4-s − 3.32·7-s + 2.23·8-s − 1.14·11-s + 1.85·13-s + 5.38·14-s − 4.85·16-s + 2.67·17-s − 1.82·19-s + 1.85·22-s − 0.420·23-s − 3.00·26-s − 2.05·28-s + 10.0·29-s − 6.00·31-s + 3.38·32-s − 4.33·34-s − 37-s + 2.94·38-s + 4.14·41-s − 8.24·43-s − 0.710·44-s + 0.679·46-s − 11.7·47-s + 4.07·49-s + 1.14·52-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s − 1.25·7-s + 0.790·8-s − 0.346·11-s + 0.515·13-s + 1.43·14-s − 1.21·16-s + 0.649·17-s − 0.417·19-s + 0.396·22-s − 0.0876·23-s − 0.589·26-s − 0.388·28-s + 1.86·29-s − 1.07·31-s + 0.597·32-s − 0.743·34-s − 0.164·37-s + 0.477·38-s + 0.646·41-s − 1.25·43-s − 0.107·44-s + 0.100·46-s − 1.70·47-s + 0.582·49-s + 0.159·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 + 3.32T + 7T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 + 1.82T + 19T^{2} \) |
| 23 | \( 1 + 0.420T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 6.00T + 31T^{2} \) |
| 41 | \( 1 - 4.14T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 - 7.07T + 59T^{2} \) |
| 61 | \( 1 + 0.294T + 61T^{2} \) |
| 67 | \( 1 - 4.44T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 9.07T + 73T^{2} \) |
| 79 | \( 1 - 9.63T + 79T^{2} \) |
| 83 | \( 1 - 0.115T + 83T^{2} \) |
| 89 | \( 1 - 5.51T + 89T^{2} \) |
| 97 | \( 1 - 1.38T + 97T^{2} \) |
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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
−7.66107929343233983899936702970, −6.74445477723103625501911985115, −6.44754080284927227106542563556, −5.44649078699030278939977928747, −4.66642021525062687487650399778, −3.70714652302711933557214901330, −3.04743478881423332314246460637, −2.00440731005532945276369167414, −0.955352044294625566107567176805, 0,
0.955352044294625566107567176805, 2.00440731005532945276369167414, 3.04743478881423332314246460637, 3.70714652302711933557214901330, 4.66642021525062687487650399778, 5.44649078699030278939977928747, 6.44754080284927227106542563556, 6.74445477723103625501911985115, 7.66107929343233983899936702970
