| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s + 2·7-s − 4·8-s − 4·10-s + 2·11-s + 6·12-s + 2·13-s − 4·14-s + 4·15-s + 5·16-s + 10·17-s − 6·19-s + 6·20-s + 4·21-s − 4·22-s + 6·23-s − 8·24-s + 3·25-s − 4·26-s − 2·27-s + 6·28-s + 16·29-s − 8·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s − 1.26·10-s + 0.603·11-s + 1.73·12-s + 0.554·13-s − 1.06·14-s + 1.03·15-s + 5/4·16-s + 2.42·17-s − 1.37·19-s + 1.34·20-s + 0.872·21-s − 0.852·22-s + 1.25·23-s − 1.63·24-s + 3/5·25-s − 0.784·26-s − 0.384·27-s + 1.13·28-s + 2.97·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.902056201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902056201\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 16 T + 119 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 16 T + 123 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 71 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 140 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 20 T + 206 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 119 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 107 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 164 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 135 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T - 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−10.99961431841786874305993870716, −10.72124447463230131315876686667, −10.50535292440714035812457313933, −9.943153749264273243336424181147, −9.309333234948764211906413474441, −9.152498231602948523777427627264, −8.566197262430930748098909763116, −8.526937701346771423689436457864, −7.79096932168984900649403720279, −7.60968912236829993084843837988, −6.76339813098567926288706145223, −6.52893355895303371305281446852, −5.65859646574230987807103901971, −5.47166556350359167234355663510, −4.54462659814139234528543846504, −3.64794824074334612399050328029, −3.00310925012367527400233058258, −2.61463251679631140882121102810, −1.59669078751643417199829952797, −1.25235821422080383469071644201,
1.25235821422080383469071644201, 1.59669078751643417199829952797, 2.61463251679631140882121102810, 3.00310925012367527400233058258, 3.64794824074334612399050328029, 4.54462659814139234528543846504, 5.47166556350359167234355663510, 5.65859646574230987807103901971, 6.52893355895303371305281446852, 6.76339813098567926288706145223, 7.60968912236829993084843837988, 7.79096932168984900649403720279, 8.526937701346771423689436457864, 8.566197262430930748098909763116, 9.152498231602948523777427627264, 9.309333234948764211906413474441, 9.943153749264273243336424181147, 10.50535292440714035812457313933, 10.72124447463230131315876686667, 10.99961431841786874305993870716
