L(s) = 1 | − 5.65·3-s + 23.0·9-s + 14i·11-s + 2·17-s + (−8.48 − 17i)19-s + 25·25-s − 79.1·27-s − 79.1i·33-s − 67.8i·41-s − 14i·43-s − 49·49-s − 11.3·51-s + (48 + 96.1i)57-s − 84.8·59-s + 118.·67-s + ⋯ |
L(s) = 1 | − 1.88·3-s + 2.55·9-s + 1.27i·11-s + 0.117·17-s + (−0.446 − 0.894i)19-s + 25-s − 2.93·27-s − 2.39i·33-s − 1.65i·41-s − 0.325i·43-s − 0.999·49-s − 0.221·51-s + (0.842 + 1.68i)57-s − 1.43·59-s + 1.77·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7279401002\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7279401002\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (8.48 + 17i)T \) |
good | 3 | \( 1 + 5.65T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 14iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 2T + 289T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 67.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 84.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 118.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 142T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 158iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 169. iT - 9.40e3T^{2} \) |
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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
−9.907917372109475480438546256607, −9.161206762609627960652475631954, −7.78035988616187501722217311177, −6.84862680085341969189925707518, −6.53565402739323325265443441333, −5.30046671552701610891103248535, −4.87819392340644531693793794419, −3.94391869203418939256021933289, −2.12808989038824131673387936688, −0.833586561754472031940426294022,
0.41043851513814025232159660260, 1.47219837268437970489084612250, 3.30906412316857354685361589623, 4.47600033779985814560226445603, 5.23899490549710441820038191632, 6.13380806634298393383302174781, 6.46130444939742382639509154723, 7.58889066872815247876012993067, 8.504909262238373238357888343434, 9.688707802584199546616321795043