| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 2·7-s + 4·8-s − 4·10-s − 2·11-s + 4·13-s + 4·14-s + 5·16-s − 2·19-s − 6·20-s − 4·22-s − 6·23-s + 3·25-s + 8·26-s + 6·28-s + 6·29-s + 4·31-s + 6·32-s − 4·35-s + 10·37-s − 4·38-s − 8·40-s + 6·41-s + 4·43-s − 6·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s + 1.41·8-s − 1.26·10-s − 0.603·11-s + 1.10·13-s + 1.06·14-s + 5/4·16-s − 0.458·19-s − 1.34·20-s − 0.852·22-s − 1.25·23-s + 3/5·25-s + 1.56·26-s + 1.13·28-s + 1.11·29-s + 0.718·31-s + 1.06·32-s − 0.676·35-s + 1.64·37-s − 0.648·38-s − 1.26·40-s + 0.937·41-s + 0.609·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.734743578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.734743578\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 96 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 180 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 310 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 22 T + 288 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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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
−7.978964998614973655197450280283, −7.75655587775575908271257549236, −7.43142441911918181066357031300, −7.14013886386978090854371717682, −6.52223053915333230002798580287, −6.24907879193331714577530983211, −5.92961118492704719945255369372, −5.85517046709022550412858726142, −5.10481920482527062188811603656, −4.82683838986969613838725365196, −4.54352712636618204622527396983, −4.24853701323207272660718394511, −3.70595806652354631805188201060, −3.70267479086179153307686302042, −2.94173907749004183753286302806, −2.71638556143364600168599886768, −2.10655789273202128722845380458, −1.87572533000356632405822892559, −0.889705387371480721795198427193, −0.75319626833221166755646784724,
0.75319626833221166755646784724, 0.889705387371480721795198427193, 1.87572533000356632405822892559, 2.10655789273202128722845380458, 2.71638556143364600168599886768, 2.94173907749004183753286302806, 3.70267479086179153307686302042, 3.70595806652354631805188201060, 4.24853701323207272660718394511, 4.54352712636618204622527396983, 4.82683838986969613838725365196, 5.10481920482527062188811603656, 5.85517046709022550412858726142, 5.92961118492704719945255369372, 6.24907879193331714577530983211, 6.52223053915333230002798580287, 7.14013886386978090854371717682, 7.43142441911918181066357031300, 7.75655587775575908271257549236, 7.978964998614973655197450280283
