| L(s) = 1 | + 4·5-s − 3·9-s + 6·11-s − 16·19-s + 11·25-s + 2·29-s + 4·31-s + 12·41-s − 12·45-s + 24·55-s − 20·59-s + 14·79-s + 16·89-s − 64·95-s − 18·99-s + 24·101-s + 14·109-s + 5·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 16·155-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 9-s + 1.80·11-s − 3.67·19-s + 11/5·25-s + 0.371·29-s + 0.718·31-s + 1.87·41-s − 1.78·45-s + 3.23·55-s − 2.60·59-s + 1.57·79-s + 1.69·89-s − 6.56·95-s − 1.80·99-s + 2.38·101-s + 1.34·109-s + 5/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.801864966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.801864966\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 185 T^{2} + p^{2} T^{4} \) |
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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
−10.41335474582043844198113590174, −9.622560021151432896492148975476, −9.308647782440550662308877980361, −8.935894304999737881634882744772, −8.839989837474824533469769095291, −8.261941328595009393414374843364, −7.85331663001520225087384247214, −7.03991450957754779616423359108, −6.48267042328027339439777400728, −6.27392318827859543239492030933, −6.17872031279806468540302374260, −5.70687099983519711550228008734, −4.92756198997538272549689692150, −4.41093566146988608438378054487, −4.18923555886163601576632259182, −3.31062373178681191280085470489, −2.71340007281772235735847648629, −2.01841951785734878764055362974, −1.86820668411512967594478598834, −0.77182193178669602856374884423,
0.77182193178669602856374884423, 1.86820668411512967594478598834, 2.01841951785734878764055362974, 2.71340007281772235735847648629, 3.31062373178681191280085470489, 4.18923555886163601576632259182, 4.41093566146988608438378054487, 4.92756198997538272549689692150, 5.70687099983519711550228008734, 6.17872031279806468540302374260, 6.27392318827859543239492030933, 6.48267042328027339439777400728, 7.03991450957754779616423359108, 7.85331663001520225087384247214, 8.261941328595009393414374843364, 8.839989837474824533469769095291, 8.935894304999737881634882744772, 9.308647782440550662308877980361, 9.622560021151432896492148975476, 10.41335474582043844198113590174
