| L(s) = 1 | − 2-s − 3-s + 6-s − 7-s + 14-s − 17-s + 21-s + 2·25-s + 32-s + 34-s − 37-s − 42-s + 2·47-s − 2·50-s + 51-s − 53-s − 59-s − 61-s − 64-s − 71-s + 74-s − 2·75-s − 79-s + 4·83-s − 89-s − 2·94-s − 96-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 6-s − 7-s + 14-s − 17-s + 21-s + 2·25-s + 32-s + 34-s − 37-s − 42-s + 2·47-s − 2·50-s + 51-s − 53-s − 59-s − 61-s − 64-s − 71-s + 74-s − 2·75-s − 79-s + 4·83-s − 89-s − 2·94-s − 96-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08912006073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08912006073\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 47 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 59 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 83 | $C_1$ | \( ( 1 - T )^{4} \) |
| 89 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 97 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
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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
−16.39761299926307357992794402644, −16.07886948847836249116832192876, −15.30703876774217792567430703967, −14.90531002592942505063566570766, −13.81816516142158278342030488727, −13.54757720598963822818329142040, −12.50277346124156882218469552755, −12.42821254608935038877218490071, −11.52817428198532142050518472672, −10.84649755632872285874832993509, −10.50369075022612179601131080455, −9.667695505824941703172707285159, −8.974116803878374901191307120128, −8.810223959306080153905100573024, −7.70850801210328648392120929641, −6.73380655474091416962152422615, −6.34108978922784854587300086860, −5.40411667131609441039217306830, −4.44331727518334815292959324093, −2.98743469578434504405692118290,
2.98743469578434504405692118290, 4.44331727518334815292959324093, 5.40411667131609441039217306830, 6.34108978922784854587300086860, 6.73380655474091416962152422615, 7.70850801210328648392120929641, 8.810223959306080153905100573024, 8.974116803878374901191307120128, 9.667695505824941703172707285159, 10.50369075022612179601131080455, 10.84649755632872285874832993509, 11.52817428198532142050518472672, 12.42821254608935038877218490071, 12.50277346124156882218469552755, 13.54757720598963822818329142040, 13.81816516142158278342030488727, 14.90531002592942505063566570766, 15.30703876774217792567430703967, 16.07886948847836249116832192876, 16.39761299926307357992794402644
