L(s) = 1 | + 14·3-s + 34·5-s + 98·7-s − 278·9-s − 420·11-s − 490·13-s + 476·15-s − 1.05e3·17-s − 1.24e3·19-s + 1.37e3·21-s + 504·23-s − 2.39e3·25-s − 7.12e3·27-s − 3.90e3·29-s − 2.04e3·31-s − 5.88e3·33-s + 3.33e3·35-s − 7.48e3·37-s − 6.86e3·39-s + 7.83e3·41-s − 1.03e4·43-s − 9.45e3·45-s − 4.19e4·47-s + 7.20e3·49-s − 1.47e4·51-s + 3.28e4·53-s − 1.42e4·55-s + ⋯ |
L(s) = 1 | + 0.898·3-s + 0.608·5-s + 0.755·7-s − 1.14·9-s − 1.04·11-s − 0.804·13-s + 0.546·15-s − 0.886·17-s − 0.791·19-s + 0.678·21-s + 0.198·23-s − 0.766·25-s − 1.88·27-s − 0.862·29-s − 0.382·31-s − 0.939·33-s + 0.459·35-s − 0.899·37-s − 0.722·39-s + 0.727·41-s − 0.852·43-s − 0.695·45-s − 2.77·47-s + 3/7·49-s − 0.795·51-s + 1.60·53-s − 0.636·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
good | 3 | $D_{4}$ | \( 1 - 14 T + 158 p T^{2} - 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 34 T + 142 p^{2} T^{2} - 34 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 420 T + 237126 T^{2} + 420 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 490 T + 656150 T^{2} + 490 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1056 T + 3106542 T^{2} + 1056 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1246 T + 1618778 T^{2} + 1246 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 504 T + 12920574 T^{2} - 504 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3904 T + 26647526 T^{2} + 3904 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2044 T + 33160782 T^{2} + 2044 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7488 T + 152693494 T^{2} + 7488 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7832 T + 168604142 T^{2} - 7832 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10332 T + 248230342 T^{2} + 10332 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 41972 T + 855029710 T^{2} + 41972 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 32812 T + 664801838 T^{2} - 32812 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 48398 T + 1851574618 T^{2} + 48398 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 718 T + 1677458142 T^{2} + 718 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12824 T + 1908078582 T^{2} + 12824 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 103992 T + 6302476942 T^{2} + 103992 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 54100 T + 3096449510 T^{2} + 54100 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 64568 T + 7121481950 T^{2} + 64568 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 47810 T + 8432588842 T^{2} - 47810 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17388 T - 1489722410 T^{2} + 17388 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 97296 T + 17237136142 T^{2} + 97296 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.07647855269186025809172753379, −10.68314610787122361674504227114, −10.14462441451551085826847267496, −9.628360931402863298881169495335, −8.938857550567959566632331451841, −8.812306754265610636529917824803, −8.143664768615809890241917948099, −7.83388689719900286671753884757, −7.25856585206097244256144596072, −6.56583246339130667938672495269, −5.69796808371529532917433742395, −5.58783773619222350943544900848, −4.77377101626411512920540647055, −4.25415424526068915276777418624, −3.17848997687018542076802660632, −2.84943190522414720071472136886, −1.91101720690821588279249872130, −1.89248897361685020734015824899, 0, 0,
1.89248897361685020734015824899, 1.91101720690821588279249872130, 2.84943190522414720071472136886, 3.17848997687018542076802660632, 4.25415424526068915276777418624, 4.77377101626411512920540647055, 5.58783773619222350943544900848, 5.69796808371529532917433742395, 6.56583246339130667938672495269, 7.25856585206097244256144596072, 7.83388689719900286671753884757, 8.143664768615809890241917948099, 8.812306754265610636529917824803, 8.938857550567959566632331451841, 9.628360931402863298881169495335, 10.14462441451551085826847267496, 10.68314610787122361674504227114, 11.07647855269186025809172753379