L(s) = 1 | − 4-s − 4·5-s − 4·11-s + 16-s − 4·19-s + 4·20-s + 11·25-s − 4·29-s − 8·31-s + 12·41-s + 4·44-s − 2·49-s + 16·55-s − 28·59-s + 20·61-s − 64-s − 16·71-s + 4·76-s + 16·79-s − 4·80-s − 36·89-s + 16·95-s − 11·100-s + 28·101-s − 24·109-s + 4·116-s − 10·121-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1.78·5-s − 1.20·11-s + 1/4·16-s − 0.917·19-s + 0.894·20-s + 11/5·25-s − 0.742·29-s − 1.43·31-s + 1.87·41-s + 0.603·44-s − 2/7·49-s + 2.15·55-s − 3.64·59-s + 2.56·61-s − 1/8·64-s − 1.89·71-s + 0.458·76-s + 1.80·79-s − 0.447·80-s − 3.81·89-s + 1.64·95-s − 1.09·100-s + 2.78·101-s − 2.29·109-s + 0.371·116-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1610874095\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1610874095\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−10.06821688937809707127692894756, −9.330818633964789614253912923314, −9.194637400753628967105753285744, −8.748423491934475628755849833790, −8.151017020020790484714756705318, −7.964217839984763862300972731444, −7.70159059975857772822177930474, −7.05738593769258433706371900339, −7.02830232270319661159796847170, −6.02489362238179160017318955529, −5.86880338955461021212004951152, −5.07101407121827046552598680767, −4.87520326704765102938360227493, −4.15677630159668975394585451273, −4.02866380992653245545035593663, −3.41919559060861455219490890985, −2.86928905207467419687942198982, −2.30146465391094410906662143949, −1.31108925300982847587427098913, −0.18476061369210921516229013215,
0.18476061369210921516229013215, 1.31108925300982847587427098913, 2.30146465391094410906662143949, 2.86928905207467419687942198982, 3.41919559060861455219490890985, 4.02866380992653245545035593663, 4.15677630159668975394585451273, 4.87520326704765102938360227493, 5.07101407121827046552598680767, 5.86880338955461021212004951152, 6.02489362238179160017318955529, 7.02830232270319661159796847170, 7.05738593769258433706371900339, 7.70159059975857772822177930474, 7.964217839984763862300972731444, 8.151017020020790484714756705318, 8.748423491934475628755849833790, 9.194637400753628967105753285744, 9.330818633964789614253912923314, 10.06821688937809707127692894756