| L(s) = 1 | − 0.924·3-s + (−0.611 − 2.15i)5-s − 1.51i·7-s − 2.14·9-s + 0.770·11-s + 4.68i·13-s + (0.564 + 1.98i)15-s + 4.00·17-s − 7.17·19-s + 1.39i·21-s + (−4.38 + 1.94i)23-s + (−4.25 + 2.62i)25-s + 4.75·27-s − 4.41·29-s − 3.07i·31-s + ⋯ |
| L(s) = 1 | − 0.533·3-s + (−0.273 − 0.961i)5-s − 0.572i·7-s − 0.715·9-s + 0.232·11-s + 1.29i·13-s + (0.145 + 0.513i)15-s + 0.972·17-s − 1.64·19-s + 0.305i·21-s + (−0.914 + 0.405i)23-s + (−0.850 + 0.525i)25-s + 0.915·27-s − 0.820·29-s − 0.551i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6603986222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6603986222\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.611 + 2.15i)T \) |
| 23 | \( 1 + (4.38 - 1.94i)T \) |
| good | 3 | \( 1 + 0.924T + 3T^{2} \) |
| 7 | \( 1 + 1.51iT - 7T^{2} \) |
| 11 | \( 1 - 0.770T + 11T^{2} \) |
| 13 | \( 1 - 4.68iT - 13T^{2} \) |
| 17 | \( 1 - 4.00T + 17T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 + 3.07iT - 31T^{2} \) |
| 37 | \( 1 - 5.52T + 37T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 - 2.03iT - 43T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 0.831iT - 59T^{2} \) |
| 61 | \( 1 - 7.18iT - 61T^{2} \) |
| 67 | \( 1 - 12.6iT - 67T^{2} \) |
| 71 | \( 1 - 2.23iT - 71T^{2} \) |
| 73 | \( 1 - 11.1iT - 73T^{2} \) |
| 79 | \( 1 - 3.98T + 79T^{2} \) |
| 83 | \( 1 - 7.33iT - 83T^{2} \) |
| 89 | \( 1 - 1.08iT - 89T^{2} \) |
| 97 | \( 1 + 2.74T + 97T^{2} \) |
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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
−9.344463929648715233863231570672, −8.619549429200214213608723313138, −7.919671241222848987274346573922, −7.01538479926102912778547625159, −6.04468994455644642239471845540, −5.48610059855835669639443838821, −4.24426089939704245584702714451, −4.00893165072450949454979950827, −2.30958480260464213774850926224, −1.04761293497457855033793881936,
0.30241789005731248106144224456, 2.23159983887845807374321441484, 3.07615431489873104351388426789, 4.01269502428069790207479797392, 5.26387450577591169682331646555, 6.00313291690552920046767967947, 6.44038635438720511543106391675, 7.65328666240280564333426061664, 8.164209285045935758487527982657, 9.053790156440977278586648633868
