| L(s) = 1 | + (3.62 + 3.72i)3-s + (10.2 + 17.8i)5-s + (18.5 + 0.432i)7-s + (−0.753 + 26.9i)9-s + (−25.9 + 44.9i)11-s + (22.2 − 38.6i)13-s + (−29.0 + 102. i)15-s + (−28.9 − 50.1i)17-s + (79.6 − 137. i)19-s + (65.4 + 70.5i)21-s + (−21.5 − 37.3i)23-s + (−148. + 257. i)25-s + (−103. + 94.9i)27-s + (−90.5 − 156. i)29-s − 5.82·31-s + ⋯ |
| L(s) = 1 | + (0.697 + 0.716i)3-s + (0.919 + 1.59i)5-s + (0.999 + 0.0233i)7-s + (−0.0279 + 0.999i)9-s + (−0.711 + 1.23i)11-s + (0.475 − 0.823i)13-s + (−0.500 + 1.76i)15-s + (−0.412 − 0.715i)17-s + (0.961 − 1.66i)19-s + (0.680 + 0.732i)21-s + (−0.195 − 0.338i)23-s + (−1.19 + 2.06i)25-s + (−0.736 + 0.676i)27-s + (−0.580 − 1.00i)29-s − 0.0337·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.836371411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.836371411\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3.62 - 3.72i)T \) |
| 7 | \( 1 + (-18.5 - 0.432i)T \) |
| good | 5 | \( 1 + (-10.2 - 17.8i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (25.9 - 44.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-22.2 + 38.6i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (28.9 + 50.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-79.6 + 137. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (21.5 + 37.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (90.5 + 156. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 5.82T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-128. + 223. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (138. - 240. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-133. - 230. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 - 81.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-94.6 - 163. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 - 125.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 379.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 790.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 64.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (187. + 324. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + 134.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-487. - 843. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-367. + 636. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-104. - 181. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.39361412692720518649367964305, −10.80229043221653493256497846497, −9.983934949595338055740798680247, −9.250324411208311957491871712964, −7.78952743481127546744428595548, −7.14419045406684249732385955614, −5.60806136950577299272951390855, −4.56971391864448819747593575546, −2.89931269993605277195253407972, −2.22818390492956743937506485729,
1.14799810926439121087725089921, 1.90535092365923231760899296681, 3.81965269433046665764943370393, 5.30669349238983493694753040449, 6.04488106990571847982048505258, 7.73072656907678845421683293203, 8.541998630423257239982505192614, 8.968125630734548282685288146906, 10.22795512297060972998068712386, 11.59917010858466581758158818143
