L(s) = 1 | + 2.46i·2-s + 1.27·3-s − 4.07·4-s + 0.595i·5-s + 3.14i·6-s − 5.10i·8-s − 1.37·9-s − 1.46·10-s − 5.30·11-s − 5.19·12-s − 0.996·13-s + 0.759i·15-s + 4.43·16-s − 1.72i·17-s − 3.38i·18-s + (−1.56 + 4.06i)19-s + ⋯ |
L(s) = 1 | + 1.74i·2-s + 0.736·3-s − 2.03·4-s + 0.266i·5-s + 1.28i·6-s − 1.80i·8-s − 0.457·9-s − 0.463·10-s − 1.59·11-s − 1.49·12-s − 0.276·13-s + 0.196i·15-s + 1.10·16-s − 0.417i·17-s − 0.797i·18-s + (−0.358 + 0.933i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0543 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0543 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.378125 - 0.358109i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.378125 - 0.358109i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 + (1.56 - 4.06i)T \) |
good | 2 | \( 1 - 2.46iT - 2T^{2} \) |
| 3 | \( 1 - 1.27T + 3T^{2} \) |
| 5 | \( 1 - 0.595iT - 5T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 13 | \( 1 + 0.996T + 13T^{2} \) |
| 17 | \( 1 + 1.72iT - 17T^{2} \) |
| 23 | \( 1 - 2.64T + 23T^{2} \) |
| 29 | \( 1 + 0.0820iT - 29T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 - 0.771iT - 37T^{2} \) |
| 41 | \( 1 + 7.80T + 41T^{2} \) |
| 43 | \( 1 + 5.74T + 43T^{2} \) |
| 47 | \( 1 - 8.93iT - 47T^{2} \) |
| 53 | \( 1 + 9.55iT - 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 2.88iT - 61T^{2} \) |
| 67 | \( 1 - 1.01iT - 67T^{2} \) |
| 71 | \( 1 - 14.9iT - 71T^{2} \) |
| 73 | \( 1 - 6.79iT - 73T^{2} \) |
| 79 | \( 1 + 11.5iT - 79T^{2} \) |
| 83 | \( 1 - 15.0iT - 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−10.36986734079395645588518592023, −9.514506507710535438008371452140, −8.582302145302478399817377570176, −8.124204864386438370051509347400, −7.39362375436127077768577347454, −6.62417300843629071940003476565, −5.48043452807319840194831253585, −5.04378859395030944971008690165, −3.62183880780860632530851494596, −2.49833744765039365837273484517,
0.20706635390275744905494709513, 1.95659187639228820200433287968, 2.77266288531446522390168239652, 3.48212395277507755174044655484, 4.74289983818796872228811305081, 5.42401147087888623064339549163, 7.12544426458171026052540995070, 8.280797339700258106202712587944, 8.782047365228813961606305022083, 9.533005334252637874076518432312