| L(s) = 1 | − 9·3-s + 11·5-s − 10·7-s + 31·9-s − 22·11-s − 22·13-s − 99·15-s − 160·17-s − 12·19-s + 90·21-s + 119·23-s − 135·25-s − 72·27-s − 138·29-s − 401·31-s + 198·33-s − 110·35-s + 341·37-s + 198·39-s − 318·41-s − 450·43-s + 341·45-s + 312·47-s + 262·49-s + 1.44e3·51-s − 400·53-s − 242·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.983·5-s − 0.539·7-s + 1.14·9-s − 0.603·11-s − 0.469·13-s − 1.70·15-s − 2.28·17-s − 0.144·19-s + 0.935·21-s + 1.07·23-s − 1.07·25-s − 0.513·27-s − 0.883·29-s − 2.32·31-s + 1.04·33-s − 0.531·35-s + 1.51·37-s + 0.812·39-s − 1.21·41-s − 1.59·43-s + 1.12·45-s + 0.968·47-s + 0.763·49-s + 3.95·51-s − 1.03·53-s − 0.593·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + p^{2} T + 50 T^{2} + p^{5} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 11 T + 256 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 10 T - 162 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 22 T + 4418 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 160 T + 15838 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 12 T + 4054 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 119 T + 15046 T^{2} - 119 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 138 T + 10762 T^{2} + 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 401 T + 3218 p T^{2} + 401 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 341 T + 117548 T^{2} - 341 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 318 T + 155266 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 450 T + 193246 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 312 T + 106270 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 400 T + 328054 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 21 T + 347794 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 462 T + 472306 T^{2} + 462 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 327 T + 409402 T^{2} + 327 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 603 T + 691270 T^{2} + 603 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 206 T + 707066 T^{2} - 206 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2026 T + 24930 p T^{2} + 2026 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1046 T + 1254046 T^{2} - 1046 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 721 T + 762904 T^{2} - 721 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1467 T + 1221460 T^{2} - 1467 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.80072996140007928203738048810, −11.43217869182875523207981948163, −11.08395063548935231372371712384, −10.60732924059941439256303126354, −10.17639037290031666053766102362, −9.512122545396612462615701465218, −9.132274925974300993556158061283, −8.649260457315214967808471289650, −7.46020362093846161507691730789, −7.28445255266712673261291904227, −6.32316700916703020744668050793, −6.20465825363728412460177000427, −5.58686854697651785974993535960, −5.08249721555386345128294042862, −4.52029281861718636569270372321, −3.55809675476593822906278627100, −2.40671834972962079853808266450, −1.75077660292205735339352452610, 0, 0,
1.75077660292205735339352452610, 2.40671834972962079853808266450, 3.55809675476593822906278627100, 4.52029281861718636569270372321, 5.08249721555386345128294042862, 5.58686854697651785974993535960, 6.20465825363728412460177000427, 6.32316700916703020744668050793, 7.28445255266712673261291904227, 7.46020362093846161507691730789, 8.649260457315214967808471289650, 9.132274925974300993556158061283, 9.512122545396612462615701465218, 10.17639037290031666053766102362, 10.60732924059941439256303126354, 11.08395063548935231372371712384, 11.43217869182875523207981948163, 11.80072996140007928203738048810
