| L(s) = 1 | + 1.41·2-s − 4.76·3-s + 2.00·4-s − 6.73·6-s + 7.05i·7-s + 2.82·8-s + 13.6·9-s − 10.4i·11-s − 9.52·12-s − 19.0·13-s + 9.98i·14-s + 4.00·16-s + 12.8i·17-s + 19.3·18-s + 22.7i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.58·3-s + 0.500·4-s − 1.12·6-s + 1.00i·7-s + 0.353·8-s + 1.52·9-s − 0.950i·11-s − 0.793·12-s − 1.46·13-s + 0.713i·14-s + 0.250·16-s + 0.754i·17-s + 1.07·18-s + 1.19i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6431750772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6431750772\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (-6.14 + 22.1i)T \) |
| good | 3 | \( 1 + 4.76T + 9T^{2} \) |
| 7 | \( 1 - 7.05iT - 49T^{2} \) |
| 11 | \( 1 + 10.4iT - 121T^{2} \) |
| 13 | \( 1 + 19.0T + 169T^{2} \) |
| 17 | \( 1 - 12.8iT - 289T^{2} \) |
| 19 | \( 1 - 22.7iT - 361T^{2} \) |
| 29 | \( 1 + 8.61T + 841T^{2} \) |
| 31 | \( 1 - 22.2T + 961T^{2} \) |
| 37 | \( 1 + 29.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 10.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 20.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 17.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 103.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 74.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 101. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 69.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 122.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 140. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 11.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 57.0iT - 9.40e3T^{2} \) |
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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
−9.621784108046492605374569266807, −8.448969002740695137985363845292, −7.47844639779256746156529624526, −6.30440599997481342214723131223, −6.00461333647608745137356523905, −5.17976393476927888106669484913, −4.49698083694622925036982183234, −3.15075404743438560103209063093, −1.86282816832609042472899540759, −0.20796130077999855392124751262,
1.09112778032654069053596937788, 2.62967303312418209345101891536, 4.13862524753187453701644866962, 4.91362941559481169979828194712, 5.26178236438941744744127370654, 6.60332213022481010342067415616, 7.04817115939557954237813551387, 7.64742055148260052536556027372, 9.445944749158452939778836908306, 10.09397019022827372245633389496
