L(s) = 1 | + 12·11-s + 8·23-s − 2·25-s + 12·29-s + 4·37-s − 20·43-s + 4·53-s − 8·67-s − 24·71-s + 8·79-s + 8·107-s − 20·109-s − 8·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 3.61·11-s + 1.66·23-s − 2/5·25-s + 2.22·29-s + 0.657·37-s − 3.04·43-s + 0.549·53-s − 0.977·67-s − 2.84·71-s + 0.900·79-s + 0.773·107-s − 1.91·109-s − 0.752·113-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.952790294\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.952790294\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 164 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
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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
−8.186606903107144253527389982554, −7.82632224636268707738844512862, −7.17671109456718182259662692951, −6.86449818490525449453570564035, −6.83599943364003155639979165473, −6.48807244167066257063252337390, −6.04388864948741487810013134854, −5.89996236354136227367083182381, −5.22714853778186210804765791226, −4.75681735409154423647814216043, −4.50341937202362248346770819380, −4.27212020799128708050598395436, −3.62953030193807424433779539844, −3.52902903902973866602280651938, −2.96865577631073130815384270789, −2.66961043669717636699923991565, −1.69263622372052197459451285730, −1.57661125791959474201251567119, −1.11981705315670534212033177055, −0.61227819689763455025954233183,
0.61227819689763455025954233183, 1.11981705315670534212033177055, 1.57661125791959474201251567119, 1.69263622372052197459451285730, 2.66961043669717636699923991565, 2.96865577631073130815384270789, 3.52902903902973866602280651938, 3.62953030193807424433779539844, 4.27212020799128708050598395436, 4.50341937202362248346770819380, 4.75681735409154423647814216043, 5.22714853778186210804765791226, 5.89996236354136227367083182381, 6.04388864948741487810013134854, 6.48807244167066257063252337390, 6.83599943364003155639979165473, 6.86449818490525449453570564035, 7.17671109456718182259662692951, 7.82632224636268707738844512862, 8.186606903107144253527389982554