| L(s) = 1 | − 1.41·2-s − 2.41·3-s + 3.41·6-s + 2.82·8-s + 2.82·9-s − 5.82·11-s − 1.58·13-s − 4.00·16-s + 5.24·17-s − 4·18-s + 6·19-s + 8.24·22-s − 4.58·23-s − 6.82·24-s + 2.24·26-s + 0.414·27-s + 2.65·29-s + 1.75·31-s + 14.0·33-s − 7.41·34-s + 6.24·37-s − 8.48·38-s + 3.82·39-s + 2.24·41-s − 2·43-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 1.39·3-s + 1.39·6-s + 0.999·8-s + 0.942·9-s − 1.75·11-s − 0.439·13-s − 1.00·16-s + 1.27·17-s − 0.942·18-s + 1.37·19-s + 1.75·22-s − 0.956·23-s − 1.39·24-s + 0.439·26-s + 0.0797·27-s + 0.493·29-s + 0.315·31-s + 2.44·33-s − 1.27·34-s + 1.02·37-s − 1.37·38-s + 0.613·39-s + 0.350·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 + 5.82T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 4.58T + 23T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 1.24T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + 0.242T + 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 4.75T + 97T^{2} \) |
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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
−9.727562904682335687954883051916, −8.351562204940317397312430777185, −7.74466456519532638811131211242, −7.07874004748954413403443473732, −5.71512687113582828835604201501, −5.32438407152611259801184244192, −4.40556925606388490498667873035, −2.78840361055997743144197075966, −1.12557278961490605786276875128, 0,
1.12557278961490605786276875128, 2.78840361055997743144197075966, 4.40556925606388490498667873035, 5.32438407152611259801184244192, 5.71512687113582828835604201501, 7.07874004748954413403443473732, 7.74466456519532638811131211242, 8.351562204940317397312430777185, 9.727562904682335687954883051916
