| L(s) = 1 | + 1.04i·3-s + 4.22i·5-s + (−2.02 − 1.69i)7-s + 1.91·9-s + (3.24 + 0.678i)11-s + 13-s − 4.39·15-s − 3.87·17-s + 7.23·19-s + (1.76 − 2.11i)21-s + 5.94·23-s − 12.8·25-s + 5.11i·27-s − 1.09i·29-s + 5.22i·31-s + ⋯ |
| L(s) = 1 | + 0.600i·3-s + 1.88i·5-s + (−0.766 − 0.641i)7-s + 0.638·9-s + (0.978 + 0.204i)11-s + 0.277·13-s − 1.13·15-s − 0.940·17-s + 1.66·19-s + (0.385 − 0.460i)21-s + 1.23·23-s − 2.57·25-s + 0.984i·27-s − 0.203i·29-s + 0.937i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.030403192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.030403192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.02 + 1.69i)T \) |
| 11 | \( 1 + (-3.24 - 0.678i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.04iT - 3T^{2} \) |
| 5 | \( 1 - 4.22iT - 5T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 - 5.94T + 23T^{2} \) |
| 29 | \( 1 + 1.09iT - 29T^{2} \) |
| 31 | \( 1 - 5.22iT - 31T^{2} \) |
| 37 | \( 1 - 3.09T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 6.88iT - 43T^{2} \) |
| 47 | \( 1 - 5.75iT - 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 + 6.74iT - 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 2.12T + 67T^{2} \) |
| 71 | \( 1 + 3.96T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 15.2iT - 79T^{2} \) |
| 83 | \( 1 + 3.04T + 83T^{2} \) |
| 89 | \( 1 + 7.88iT - 89T^{2} \) |
| 97 | \( 1 + 7.43iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.031871561818420872783355012258, −7.60367277871085773364043638905, −7.11320912513920712218658219604, −6.68191349022423743591605200485, −6.01278246018226384290018294763, −4.78107773849134028773353719070, −3.89699891766701182063377144415, −3.37719464665834990311374186464, −2.67426148334788684305219060297, −1.24146009600759958793894525056,
0.69926668911477828928494533014, 1.32059301609471223994593578224, 2.39935315187007193909201459501, 3.72699702929425986469332645420, 4.36892360646885357035070473122, 5.34210411532086999539854304602, 5.82111179902390761138007098593, 6.83013922107648937472632679190, 7.35199074708194477953040129163, 8.490462161202244286130854533786
