| L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 3·9-s + 11-s + 2·13-s + 4·15-s − 3·17-s + 6·19-s − 2·21-s − 9·23-s + 3·25-s − 4·27-s − 10·29-s + 4·31-s − 2·33-s − 2·35-s + 37-s − 4·39-s − 7·41-s + 10·43-s − 6·45-s − 16·47-s − 9·49-s + 6·51-s − 3·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 9-s + 0.301·11-s + 0.554·13-s + 1.03·15-s − 0.727·17-s + 1.37·19-s − 0.436·21-s − 1.87·23-s + 3/5·25-s − 0.769·27-s − 1.85·29-s + 0.718·31-s − 0.348·33-s − 0.338·35-s + 0.164·37-s − 0.640·39-s − 1.09·41-s + 1.52·43-s − 0.894·45-s − 2.33·47-s − 9/7·49-s + 0.840·51-s − 0.412·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 56 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 104 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 96 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 15 T + 210 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 180 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−7.82038171215026087245501654068, −7.65856100395173565515156813668, −7.02125006164078509905190075154, −6.85166304160319166402131679070, −6.35950954541774044255078532498, −6.24764383319733267633620885611, −5.54433546590170769430263092655, −5.54415433676742481001784086939, −4.87992443517419836727323387802, −4.80722654711593758412274821059, −4.07301754445164369321147418371, −4.03470827416950096689843433834, −3.46554223239343059173421987636, −3.28589785160579958631667920412, −2.37749325887620567266982899958, −2.04412758580748927377093516326, −1.34546198558364320856407430093, −1.09168530388002531156566765696, 0, 0,
1.09168530388002531156566765696, 1.34546198558364320856407430093, 2.04412758580748927377093516326, 2.37749325887620567266982899958, 3.28589785160579958631667920412, 3.46554223239343059173421987636, 4.03470827416950096689843433834, 4.07301754445164369321147418371, 4.80722654711593758412274821059, 4.87992443517419836727323387802, 5.54415433676742481001784086939, 5.54433546590170769430263092655, 6.24764383319733267633620885611, 6.35950954541774044255078532498, 6.85166304160319166402131679070, 7.02125006164078509905190075154, 7.65856100395173565515156813668, 7.82038171215026087245501654068
