| L(s) = 1 | − 4·5-s − 14·7-s + 36·11-s − 12·13-s − 44·17-s − 8·19-s + 76·23-s − 118·25-s − 160·29-s + 88·31-s + 56·35-s + 180·37-s − 356·41-s + 56·43-s + 616·47-s + 147·49-s − 808·53-s − 144·55-s + 504·59-s + 572·61-s + 48·65-s + 504·67-s + 1.18e3·71-s + 572·73-s − 504·77-s + 784·79-s + 1.34e3·83-s + ⋯ |
| L(s) = 1 | − 0.357·5-s − 0.755·7-s + 0.986·11-s − 0.256·13-s − 0.627·17-s − 0.0965·19-s + 0.689·23-s − 0.943·25-s − 1.02·29-s + 0.509·31-s + 0.270·35-s + 0.799·37-s − 1.35·41-s + 0.198·43-s + 1.91·47-s + 3/7·49-s − 2.09·53-s − 0.353·55-s + 1.11·59-s + 1.20·61-s + 0.0915·65-s + 0.919·67-s + 1.98·71-s + 0.917·73-s − 0.745·77-s + 1.11·79-s + 1.77·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.445939477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.445939477\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 36 T + 1906 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 12 T + 2510 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 44 T + 9230 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 11814 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 76 T + 22778 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 160 T + 53258 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 88 T + 59598 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 180 T + 15326 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 356 T + 159806 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 56 T + 36918 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 616 T + 244430 T^{2} - 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 808 T + 422090 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 504 T + 416182 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 572 T + 533838 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 504 T + 473030 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1188 T + 1068538 T^{2} - 1188 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 572 T + 237750 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 784 T + 517662 T^{2} - 784 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1344 T + 1564438 T^{2} - 1344 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 308 T + 987134 T^{2} + 308 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1052 T + 1172742 T^{2} - 1052 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.563131402391126478674858324415, −9.497210747570914165001953008685, −9.060800452255523506296312953939, −8.619022127774448573810140355528, −7.998400705858997468001251699826, −7.86065354682850525438108250898, −7.09347178952635585030051104891, −6.91613454823110207827876837448, −6.29487730268686265618310314440, −6.24202964002653225172935798224, −5.32093838882375431080248953328, −5.18295009730845738897998391215, −4.30784930554455216114642184446, −4.07653085500289626314649539550, −3.47331768844990555428260037802, −3.16828713399230423523443104578, −2.13239528938800475873200637958, −2.06701139303088339342206698395, −0.868288497165600622276958779322, −0.50546125689132150946311443445,
0.50546125689132150946311443445, 0.868288497165600622276958779322, 2.06701139303088339342206698395, 2.13239528938800475873200637958, 3.16828713399230423523443104578, 3.47331768844990555428260037802, 4.07653085500289626314649539550, 4.30784930554455216114642184446, 5.18295009730845738897998391215, 5.32093838882375431080248953328, 6.24202964002653225172935798224, 6.29487730268686265618310314440, 6.91613454823110207827876837448, 7.09347178952635585030051104891, 7.86065354682850525438108250898, 7.998400705858997468001251699826, 8.619022127774448573810140355528, 9.060800452255523506296312953939, 9.497210747570914165001953008685, 9.563131402391126478674858324415
