| L(s) = 1 | + 4·2-s − 6·3-s + 12·4-s − 10·5-s − 24·6-s + 24·7-s + 32·8-s + 27·9-s − 40·10-s − 20·11-s − 72·12-s − 20·13-s + 96·14-s + 60·15-s + 80·16-s + 108·18-s + 38·19-s − 120·20-s − 144·21-s − 80·22-s + 212·23-s − 192·24-s + 75·25-s − 80·26-s − 108·27-s + 288·28-s + 216·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 1.29·7-s + 1.41·8-s + 9-s − 1.26·10-s − 0.548·11-s − 1.73·12-s − 0.426·13-s + 1.83·14-s + 1.03·15-s + 5/4·16-s + 1.41·18-s + 0.458·19-s − 1.34·20-s − 1.49·21-s − 0.775·22-s + 1.92·23-s − 1.63·24-s + 3/5·25-s − 0.603·26-s − 0.769·27-s + 1.94·28-s + 1.38·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.371528822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.371528822\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 24 T + 724 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 20 T + 112 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 20 T + 3540 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6010 T^{2} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 212 T + 35146 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 216 T + 36592 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 60 T + 9178 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 84 T + 46996 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 504 T + 192760 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 368 T + 162236 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 444 T + 222586 T^{2} - 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 304 T + 320434 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1176 T + 752686 T^{2} - 1176 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 344 T + 481850 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 32 T + 432182 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 24 p T + 1346326 T^{2} - 24 p^{4} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1876 T + 1642614 T^{2} - 1876 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 48 T + 944254 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 724 T + 1272922 T^{2} - 724 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 184 T - 456632 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1956 T + 2547676 T^{2} + 1956 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.79089935184641325957243013454, −10.70572714969131225572640054565, −9.696115709735650030307792791899, −9.499663707962585355977698344058, −8.468176966005248031865593014837, −8.276398052806071520099941318708, −7.57099321971451522738370418252, −7.38068892458143019267502011460, −6.72518728796162156559885410961, −6.60371393829504087446896627888, −5.52145652580856177449476077658, −5.44106474351794299410342844310, −4.95378986618413356273649948996, −4.66104196589423113526566456933, −3.97913211814469456250018686799, −3.66533815710626663265942849882, −2.58179951728695880505309285214, −2.35626727105036704795230754720, −1.00149348271823526281186398153, −0.839590868171534499030550963315,
0.839590868171534499030550963315, 1.00149348271823526281186398153, 2.35626727105036704795230754720, 2.58179951728695880505309285214, 3.66533815710626663265942849882, 3.97913211814469456250018686799, 4.66104196589423113526566456933, 4.95378986618413356273649948996, 5.44106474351794299410342844310, 5.52145652580856177449476077658, 6.60371393829504087446896627888, 6.72518728796162156559885410961, 7.38068892458143019267502011460, 7.57099321971451522738370418252, 8.276398052806071520099941318708, 8.468176966005248031865593014837, 9.499663707962585355977698344058, 9.696115709735650030307792791899, 10.70572714969131225572640054565, 10.79089935184641325957243013454
