| L(s) = 1 | + 100·5-s + 1.44e3·9-s + 4.48e3·13-s + 5.79e3·17-s − 2.37e4·25-s − 4.19e4·29-s + 804·37-s + 2.66e4·41-s + 1.44e5·45-s + 2.01e5·49-s + 3.43e5·53-s − 5.41e5·61-s + 4.48e5·65-s − 1.39e6·73-s + 1.54e6·81-s + 5.79e5·85-s − 3.23e5·89-s + 1.04e6·97-s + 3.19e6·101-s − 3.92e6·109-s + 3.53e6·113-s + 6.46e6·117-s − 9.00e5·121-s − 4.18e6·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 4/5·5-s + 1.97·9-s + 2.04·13-s + 1.17·17-s − 1.51·25-s − 1.72·29-s + 0.0158·37-s + 0.386·41-s + 1.58·45-s + 1.71·49-s + 2.30·53-s − 2.38·61-s + 1.63·65-s − 3.58·73-s + 2.91·81-s + 0.943·85-s − 0.458·89-s + 1.14·97-s + 3.10·101-s − 3.02·109-s + 2.44·113-s + 4.03·117-s − 0.508·121-s − 2.14·125-s − 1.37·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.356723435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.356723435\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 1442 T^{2} + p^{12} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 p^{2} T + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 201442 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 900542 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2242 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2898 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 29257058 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 93182 p^{2} T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 20990 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 422727038 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 402 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 13330 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12606965698 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 596024002 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 171570 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 62659063426 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 270878 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 5372154206 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 254706899938 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 696606 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 37146261826 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 563099056738 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 161598 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 520306 T + p^{6} T^{2} )^{2} \) |
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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
−15.58884467137024277747438973154, −15.54426792500702816774303018506, −14.62744028540702363745568140669, −13.75616670964216313001903572232, −13.30718945497033715327020771938, −13.08226510518949671167985555966, −12.14458956504293651085469771784, −11.50485667880352880372542441103, −10.38452849788969214258603197907, −10.33717385192816846597432225247, −9.419001990250780307957037133171, −8.854209415263838794912312828513, −7.67865485373985707702198535417, −7.25868589340695756920027144392, −6.03032228933356484082202579996, −5.71324254821469090649438743081, −4.23420152532468371984594567931, −3.62455928271443368042648450628, −1.84655084650369656166958470320, −1.15574642889432793977596692190,
1.15574642889432793977596692190, 1.84655084650369656166958470320, 3.62455928271443368042648450628, 4.23420152532468371984594567931, 5.71324254821469090649438743081, 6.03032228933356484082202579996, 7.25868589340695756920027144392, 7.67865485373985707702198535417, 8.854209415263838794912312828513, 9.419001990250780307957037133171, 10.33717385192816846597432225247, 10.38452849788969214258603197907, 11.50485667880352880372542441103, 12.14458956504293651085469771784, 13.08226510518949671167985555966, 13.30718945497033715327020771938, 13.75616670964216313001903572232, 14.62744028540702363745568140669, 15.54426792500702816774303018506, 15.58884467137024277747438973154
