| L(s) = 1 | − 2.37·2-s + 3.66·4-s + 2.28·7-s − 3.95·8-s − 1.12·11-s + 5.95·13-s − 5.43·14-s + 2.09·16-s − 5.80·17-s − 4.08·19-s + 2.67·22-s + 23-s − 14.1·26-s + 8.36·28-s + 0.408·29-s − 3.19·31-s + 2.93·32-s + 13.8·34-s − 9.80·37-s + 9.71·38-s − 6.27·41-s + 7.75·43-s − 4.11·44-s − 2.37·46-s + 6.40·47-s − 1.78·49-s + 21.8·52-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.83·4-s + 0.863·7-s − 1.39·8-s − 0.338·11-s + 1.65·13-s − 1.45·14-s + 0.523·16-s − 1.40·17-s − 0.936·19-s + 0.570·22-s + 0.208·23-s − 2.78·26-s + 1.58·28-s + 0.0758·29-s − 0.573·31-s + 0.518·32-s + 2.36·34-s − 1.61·37-s + 1.57·38-s − 0.979·41-s + 1.18·43-s − 0.621·44-s − 0.350·46-s + 0.933·47-s − 0.254·49-s + 3.02·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8358875048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8358875048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 29 | \( 1 - 0.408T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 + 6.27T + 41T^{2} \) |
| 43 | \( 1 - 7.75T + 43T^{2} \) |
| 47 | \( 1 - 6.40T + 47T^{2} \) |
| 53 | \( 1 - 6.73T + 53T^{2} \) |
| 59 | \( 1 - 4.75T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 - 0.283T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 9.61T + 73T^{2} \) |
| 79 | \( 1 - 4.48T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.418508633181532903058694705926, −7.80118150403454085476016575466, −6.90292070090434459778213095660, −6.45630136186613921094117523237, −5.49049412356521712456368105459, −4.50146902169841718255542695995, −3.59525755909214843009197644573, −2.26558935105830000294267645401, −1.73593814318007764445385673273, −0.64790115211806495739304321589,
0.64790115211806495739304321589, 1.73593814318007764445385673273, 2.26558935105830000294267645401, 3.59525755909214843009197644573, 4.50146902169841718255542695995, 5.49049412356521712456368105459, 6.45630136186613921094117523237, 6.90292070090434459778213095660, 7.80118150403454085476016575466, 8.418508633181532903058694705926
