| L(s) = 1 | + (−1.41 + 1.41i)5-s + 7-s + (0.748 + 0.748i)11-s + (4.24 − 4.24i)13-s − 7.50i·17-s − 4.59i·23-s + 0.999i·25-s + (−5.26 − 5.26i)29-s + 8.49i·31-s + (−1.41 + 1.41i)35-s + (−1 − i)37-s − 0.978·41-s + (−6.80 + 6.80i)43-s − 6.36·47-s + 49-s + ⋯ |
| L(s) = 1 | + (−0.632 + 0.632i)5-s + 0.377·7-s + (0.225 + 0.225i)11-s + (1.17 − 1.17i)13-s − 1.82i·17-s − 0.958i·23-s + 0.199i·25-s + (−0.976 − 0.976i)29-s + 1.52i·31-s + (−0.239 + 0.239i)35-s + (−0.164 − 0.164i)37-s − 0.152·41-s + (−1.03 + 1.03i)43-s − 0.927·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.154629891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.154629891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + (1.41 - 1.41i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.748 - 0.748i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.24 + 4.24i)T - 13iT^{2} \) |
| 17 | \( 1 + 7.50iT - 17T^{2} \) |
| 19 | \( 1 + 19iT^{2} \) |
| 23 | \( 1 + 4.59iT - 23T^{2} \) |
| 29 | \( 1 + (5.26 + 5.26i)T + 29iT^{2} \) |
| 31 | \( 1 - 8.49iT - 31T^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.978T + 41T^{2} \) |
| 43 | \( 1 + (6.80 - 6.80i)T - 43iT^{2} \) |
| 47 | \( 1 + 6.36T + 47T^{2} \) |
| 53 | \( 1 + (8.17 - 8.17i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.51 + 4.51i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.249 + 0.249i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.750 + 0.750i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.62iT - 71T^{2} \) |
| 73 | \( 1 + 8.61iT - 73T^{2} \) |
| 79 | \( 1 + 16.9iT - 79T^{2} \) |
| 83 | \( 1 + (-2.47 + 2.47i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.55T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−8.022739902469615165453920430484, −7.65568413861854281979674725662, −6.80492344992524291561662901624, −6.14601683917979694037128661238, −5.14447780439599350940589156275, −4.51926678507128309012271019894, −3.33252764619169513833162443669, −3.02621976412094993870146414844, −1.62660643205833063534009967752, −0.34548316793586413472448547926,
1.28996803532622112715898430686, 1.94301065124882894089007013742, 3.75418869249989470797844117919, 3.77804150406517387296013055971, 4.80673655906534576730173393980, 5.73392161303265346458557861118, 6.38618633263167738020541776755, 7.21746197625395856805965877460, 8.249765451644898021848828060552, 8.404072058556049911451541207444
