L(s) = 1 | − 1.41i·2-s − 2.41i·3-s − 3.41·6-s − 2.82i·8-s − 2.82·9-s − 5.82·11-s − 1.58i·13-s − 4.00·16-s − 5.24i·17-s + 4i·18-s + 6·19-s + 8.24i·22-s + 4.58i·23-s − 6.82·24-s − 2.24·26-s − 0.414i·27-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.39i·3-s − 1.39·6-s − 0.999i·8-s − 0.942·9-s − 1.75·11-s − 0.439i·13-s − 1.00·16-s − 1.27i·17-s + 0.942i·18-s + 1.37·19-s + 1.75i·22-s + 0.956i·23-s − 1.39·24-s − 0.439·26-s − 0.0797i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.270900893\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.270900893\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.41iT - 2T^{2} \) |
| 3 | \( 1 + 2.41iT - 3T^{2} \) |
| 11 | \( 1 + 5.82T + 11T^{2} \) |
| 13 | \( 1 + 1.58iT - 13T^{2} \) |
| 17 | \( 1 + 5.24iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 4.58iT - 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 - 6.24iT - 37T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 1.24iT - 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 + 0.242iT - 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 + 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 4.75iT - 97T^{2} \) |
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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
−9.437756366726304412357360664988, −8.085336882824043203384891624261, −7.48267745081175956855110392817, −6.98796468974389502323412828151, −5.78928336388142164310111430713, −4.97473790733920081220180753826, −3.25529125406817869078892988610, −2.68184558177200711914750519250, −1.65146466393061523194375995608, −0.50832121335449841800534236296,
2.28259666147370226184679726090, 3.43116280685049105229478629439, 4.53872707093786199692961572561, 5.33421194560388637154149162104, 5.84192138597272278981041339639, 7.05450563969276747852234438465, 7.82631391921602856553650286777, 8.561837369064377816117266735055, 9.389154250894299167631232059919, 10.38718853870190943045447159181