| L(s) = 1 | − 4·5-s + 4·11-s − 2·13-s + 8·17-s − 8·19-s − 4·23-s + 5·25-s − 12·29-s + 8·31-s − 6·37-s − 16·43-s − 2·49-s + 8·53-s − 16·55-s + 16·59-s − 14·61-s + 8·65-s − 8·67-s + 4·71-s + 2·73-s − 8·79-s + 8·83-s − 32·85-s + 32·95-s + 4·97-s + 24·101-s − 8·103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.20·11-s − 0.554·13-s + 1.94·17-s − 1.83·19-s − 0.834·23-s + 25-s − 2.22·29-s + 1.43·31-s − 0.986·37-s − 2.43·43-s − 2/7·49-s + 1.09·53-s − 2.15·55-s + 2.08·59-s − 1.79·61-s + 0.992·65-s − 0.977·67-s + 0.474·71-s + 0.234·73-s − 0.900·79-s + 0.878·83-s − 3.47·85-s + 3.28·95-s + 0.406·97-s + 2.38·101-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 159 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 134 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.963191754423255191041969096621, −7.69040914158382638692131844503, −7.48775744853638712105856991126, −6.98228257023412052865021657583, −6.62093962551675721140826533148, −6.39694576065037735983292923065, −5.85997307360459704868112536852, −5.47908510744616675049283055097, −5.06111842535546843917427456456, −4.63437514986110252213669112206, −4.07211275064236649378924044750, −3.99725712113033523917867295831, −3.47607333494340972460268235153, −3.44857663683522101101179363398, −2.67600746142326620992334463623, −2.07782934499415233193912037356, −1.60675655354529129130464912780, −1.04613774972967466695169698641, 0, 0,
1.04613774972967466695169698641, 1.60675655354529129130464912780, 2.07782934499415233193912037356, 2.67600746142326620992334463623, 3.44857663683522101101179363398, 3.47607333494340972460268235153, 3.99725712113033523917867295831, 4.07211275064236649378924044750, 4.63437514986110252213669112206, 5.06111842535546843917427456456, 5.47908510744616675049283055097, 5.85997307360459704868112536852, 6.39694576065037735983292923065, 6.62093962551675721140826533148, 6.98228257023412052865021657583, 7.48775744853638712105856991126, 7.69040914158382638692131844503, 7.963191754423255191041969096621
