L(s) = 1 | + (1.27 − 0.817i)3-s + (−0.415 − 0.909i)5-s + (0.540 − 0.158i)7-s + (0.533 − 1.16i)9-s + (−1.27 − 0.817i)15-s + (0.557 − 0.643i)21-s + (−0.755 − 0.654i)23-s + (−0.654 + 0.755i)25-s + (−0.0612 − 0.425i)27-s + (−0.118 + 0.822i)29-s + (−0.368 − 0.425i)35-s + (0.797 + 1.74i)41-s + (1.66 − 1.07i)43-s − 1.28·45-s − 1.81·47-s + ⋯ |
L(s) = 1 | + (1.27 − 0.817i)3-s + (−0.415 − 0.909i)5-s + (0.540 − 0.158i)7-s + (0.533 − 1.16i)9-s + (−1.27 − 0.817i)15-s + (0.557 − 0.643i)21-s + (−0.755 − 0.654i)23-s + (−0.654 + 0.755i)25-s + (−0.0612 − 0.425i)27-s + (−0.118 + 0.822i)29-s + (−0.368 − 0.425i)35-s + (0.797 + 1.74i)41-s + (1.66 − 1.07i)43-s − 1.28·45-s − 1.81·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.678150349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.678150349\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (0.755 + 0.654i)T \) |
good | 3 | \( 1 + (-1.27 + 0.817i)T + (0.415 - 0.909i)T^{2} \) |
| 7 | \( 1 + (-0.540 + 0.158i)T + (0.841 - 0.540i)T^{2} \) |
| 11 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 13 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 17 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 29 | \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \) |
| 31 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 37 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 41 | \( 1 + (-0.797 - 1.74i)T + (-0.654 + 0.755i)T^{2} \) |
| 43 | \( 1 + (-1.66 + 1.07i)T + (0.415 - 0.909i)T^{2} \) |
| 47 | \( 1 + 1.81T + T^{2} \) |
| 53 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 67 | \( 1 + (-0.708 + 0.817i)T + (-0.142 - 0.989i)T^{2} \) |
| 71 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 73 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 83 | \( 1 + (-0.449 + 0.983i)T + (-0.654 - 0.755i)T^{2} \) |
| 89 | \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \) |
| 97 | \( 1 + (0.654 - 0.755i)T^{2} \) |
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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
−9.043883938321858802765413044862, −8.288930650256984835444554117776, −7.934748929374722267046294856215, −7.21596307700679144244450240558, −6.21528557187779552835220347364, −5.03911834197044685804610161347, −4.21366389679827176302665879657, −3.25369728798643984754011083655, −2.14372405106164536355858951246, −1.21124367485581951808216410072,
2.05851927808883723713675847581, 2.86633313707992347076146144957, 3.78089161477507288015769524297, 4.32540065371331391599371973365, 5.50965618437103400294812187150, 6.55039614734892784265011767547, 7.66265062139495134630054744531, 7.980596711122766750876182091820, 8.841132390356708905920538157919, 9.645585811059878048730036040836