| L(s) = 1 | − 3-s + 10·5-s − 57·7-s − 9·9-s − 10·11-s + 13·13-s − 10·15-s − 51·17-s + 38·19-s + 57·21-s + 155·23-s + 2·25-s − 8·27-s − 79·29-s + 16·31-s + 10·33-s − 570·35-s + 380·37-s − 13·39-s − 790·41-s − 296·43-s − 90·45-s + 200·47-s + 1.79e3·49-s + 51·51-s + 397·53-s − 100·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 0.894·5-s − 3.07·7-s − 1/3·9-s − 0.274·11-s + 0.277·13-s − 0.172·15-s − 0.727·17-s + 0.458·19-s + 0.592·21-s + 1.40·23-s + 0.0159·25-s − 0.0570·27-s − 0.505·29-s + 0.0926·31-s + 0.0527·33-s − 2.75·35-s + 1.68·37-s − 0.0533·39-s − 3.00·41-s − 1.04·43-s − 0.298·45-s + 0.620·47-s + 5.23·49-s + 0.140·51-s + 1.02·53-s − 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3950417103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3950417103\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 10 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 p T + 98 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 57 T + 1454 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 10 T + 2510 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - p T + 2268 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 p T + 520 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 155 T + 994 p T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 79 T + 13124 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 16 T + 48318 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 380 T + 126078 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 790 T + 292274 T^{2} + 790 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 296 T + 78966 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 200 T + 146846 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 397 T + 333572 T^{2} - 397 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 201 T + 197794 T^{2} + 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 680 T + 483894 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 939 T + 740138 T^{2} - 939 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 406 T + 735614 T^{2} + 406 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 123 T + 781772 T^{2} - 123 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 106 T + 840030 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2226 T + 2380750 T^{2} + 2226 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 870 T + 1594738 T^{2} + 870 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1864 T + 2438382 T^{2} + 1864 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.46303202895333273951763670784, −11.05910987873668728918799327363, −10.34941093980517468289343113719, −9.981163172924200339816738226491, −9.753019332548844287154885992461, −9.309943672209246386681113718483, −8.848055249909092413619455276639, −8.424244474252052420686223716343, −7.43991704410451753645865133801, −6.89358553620501172039810346309, −6.56371704044515777332019170136, −6.26798229834100459760831554684, −5.53444048974252642930265960347, −5.33340538823339034327753409774, −4.21572117142902234421037269287, −3.57295554647611239972677773037, −2.81470004419783778171515280995, −2.78061474172021342179272012302, −1.43933601745013891677966158617, −0.23359380637656218443573556698,
0.23359380637656218443573556698, 1.43933601745013891677966158617, 2.78061474172021342179272012302, 2.81470004419783778171515280995, 3.57295554647611239972677773037, 4.21572117142902234421037269287, 5.33340538823339034327753409774, 5.53444048974252642930265960347, 6.26798229834100459760831554684, 6.56371704044515777332019170136, 6.89358553620501172039810346309, 7.43991704410451753645865133801, 8.424244474252052420686223716343, 8.848055249909092413619455276639, 9.309943672209246386681113718483, 9.753019332548844287154885992461, 9.981163172924200339816738226491, 10.34941093980517468289343113719, 11.05910987873668728918799327363, 11.46303202895333273951763670784
