L(s) = 1 | − 2-s + (−1.28 − 1.28i)3-s + 4-s + (1.63 − 1.52i)5-s + (1.28 + 1.28i)6-s + (−3.01 − 3.01i)7-s − 8-s + 0.323i·9-s + (−1.63 + 1.52i)10-s + 2.38i·11-s + (−1.28 − 1.28i)12-s + 1.44·13-s + (3.01 + 3.01i)14-s + (−4.07 − 0.148i)15-s + 16-s − 2.09i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.744 − 0.744i)3-s + 0.5·4-s + (0.732 − 0.680i)5-s + (0.526 + 0.526i)6-s + (−1.13 − 1.13i)7-s − 0.353·8-s + 0.107i·9-s + (−0.517 + 0.481i)10-s + 0.720i·11-s + (−0.372 − 0.372i)12-s + 0.401·13-s + (0.805 + 0.805i)14-s + (−1.05 − 0.0384i)15-s + 0.250·16-s − 0.508i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0287543 - 0.487122i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0287543 - 0.487122i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (-1.63 + 1.52i)T \) |
| 37 | \( 1 + (1.89 - 5.78i)T \) |
good | 3 | \( 1 + (1.28 + 1.28i)T + 3iT^{2} \) |
| 7 | \( 1 + (3.01 + 3.01i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.38iT - 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 + 2.09iT - 17T^{2} \) |
| 19 | \( 1 + (2.46 - 2.46i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.168T + 23T^{2} \) |
| 29 | \( 1 + (4.74 + 4.74i)T + 29iT^{2} \) |
| 31 | \( 1 + (3.34 - 3.34i)T - 31iT^{2} \) |
| 41 | \( 1 + 6.22iT - 41T^{2} \) |
| 43 | \( 1 + 3.07T + 43T^{2} \) |
| 47 | \( 1 + (4.67 + 4.67i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.87 - 2.87i)T - 53iT^{2} \) |
| 59 | \( 1 + (-6.78 + 6.78i)T - 59iT^{2} \) |
| 61 | \( 1 + (-6.94 + 6.94i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7.89 + 7.89i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + (-4.83 - 4.83i)T + 73iT^{2} \) |
| 79 | \( 1 + (5.31 - 5.31i)T - 79iT^{2} \) |
| 83 | \( 1 + (-8.43 + 8.43i)T - 83iT^{2} \) |
| 89 | \( 1 + (11.6 + 11.6i)T + 89iT^{2} \) |
| 97 | \( 1 - 5.04iT - 97T^{2} \) |
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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
−10.81499756231601028421522873400, −9.903262935357367270674746772401, −9.391317234961806335481965446352, −8.087816504146326162100051505116, −6.89367623060756308030117597941, −6.51447615498759902722389256284, −5.36693285188112948028517390926, −3.74699968918432392976650693774, −1.78589529398794047180456442993, −0.43459862122650478939976492554,
2.31616812252778997381831312401, 3.53985366745794822970880741945, 5.46584747974478922396777299584, 6.00909457227823162922520337482, 6.86059249979427211175617853435, 8.436887787965877128561640752249, 9.349152029015375479298843832931, 9.937847353555167596249882138372, 10.93440365544469145184947695348, 11.29573403388648904768073967404