L(s) = 1 | − 5-s − 4·17-s + 4·19-s − 6·23-s + 25-s + 2·29-s − 6·37-s − 10·41-s − 4·43-s − 10·47-s − 7·49-s − 2·53-s + 4·59-s + 14·61-s + 2·67-s − 4·71-s + 4·73-s + 8·79-s + 12·83-s + 4·85-s − 6·89-s − 4·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.970·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.45·47-s − 49-s − 0.274·53-s + 0.520·59-s + 1.79·61-s + 0.244·67-s − 0.474·71-s + 0.468·73-s + 0.900·79-s + 1.31·83-s + 0.433·85-s − 0.635·89-s − 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.210347897\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.210347897\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.74876910872904, −15.00325578603350, −14.48909354455482, −13.93918630620643, −13.33162008326960, −12.96452572274395, −12.03701087032015, −11.84010246989381, −11.26898813833976, −10.59925921296287, −9.978859687843149, −9.562355664999373, −8.741271892772494, −8.235364278494929, −7.827139537600140, −6.808360016558293, −6.732257755695266, −5.794198301743924, −5.035637840205075, −4.620972827652827, −3.641945660505374, −3.334017110607215, −2.261793318634961, −1.612148419931644, −0.4394188709132055,
0.4394188709132055, 1.612148419931644, 2.261793318634961, 3.334017110607215, 3.641945660505374, 4.620972827652827, 5.035637840205075, 5.794198301743924, 6.732257755695266, 6.808360016558293, 7.827139537600140, 8.235364278494929, 8.741271892772494, 9.562355664999373, 9.978859687843149, 10.59925921296287, 11.26898813833976, 11.84010246989381, 12.03701087032015, 12.96452572274395, 13.33162008326960, 13.93918630620643, 14.48909354455482, 15.00325578603350, 15.74876910872904