| L(s) = 1 | − 2·2-s + 3·3-s + 3·4-s − 4·5-s − 6·6-s − 2·7-s − 4·8-s + 2·9-s + 8·10-s + 6·11-s + 9·12-s − 13-s + 4·14-s − 12·15-s + 5·16-s − 3·17-s − 4·18-s − 12·20-s − 6·21-s − 12·22-s − 12·24-s + 7·25-s + 2·26-s − 6·27-s − 6·28-s − 12·29-s + 24·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 3/2·4-s − 1.78·5-s − 2.44·6-s − 0.755·7-s − 1.41·8-s + 2/3·9-s + 2.52·10-s + 1.80·11-s + 2.59·12-s − 0.277·13-s + 1.06·14-s − 3.09·15-s + 5/4·16-s − 0.727·17-s − 0.942·18-s − 2.68·20-s − 1.30·21-s − 2.55·22-s − 2.44·24-s + 7/5·25-s + 0.392·26-s − 1.15·27-s − 1.13·28-s − 2.22·29-s + 4.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5860242882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5860242882\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 53 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 3 p T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 77 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 11 T + 63 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 127 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 27 T + 329 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 183 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 293 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−8.362040161291152178920979262415, −8.191891778338104501150534830947, −7.77220179263387403117666539294, −7.72758387381972566852892755863, −7.01120316441412238598491720538, −6.85680439886298026459382448769, −6.50042864542864021119050423429, −6.36794444436543299655366913367, −5.40533759026040380016170167183, −5.30194560903042443945436817170, −4.35823881154476914691464121369, −4.07520709882820262496942915551, −3.66612959596430928730819504198, −3.50298659509189696801340096952, −3.07755957027637096993872645079, −2.66545799586341040005634387415, −2.02620141671602050552620881403, −1.79116867772539370725798303792, −0.950268655545674268484101109718, −0.26849653969629374787576615987,
0.26849653969629374787576615987, 0.950268655545674268484101109718, 1.79116867772539370725798303792, 2.02620141671602050552620881403, 2.66545799586341040005634387415, 3.07755957027637096993872645079, 3.50298659509189696801340096952, 3.66612959596430928730819504198, 4.07520709882820262496942915551, 4.35823881154476914691464121369, 5.30194560903042443945436817170, 5.40533759026040380016170167183, 6.36794444436543299655366913367, 6.50042864542864021119050423429, 6.85680439886298026459382448769, 7.01120316441412238598491720538, 7.72758387381972566852892755863, 7.77220179263387403117666539294, 8.191891778338104501150534830947, 8.362040161291152178920979262415
