| L(s) = 1 | + 5-s + (−2.27 − 1.35i)7-s − 2.71i·11-s + 6.54i·13-s − 1.53·17-s − 2.30i·19-s + 3.83i·23-s + 25-s − 3.83i·29-s − 3.25i·31-s + (−2.27 − 1.35i)35-s + 3.01·37-s + 6.54·41-s + 0.468·43-s − 9.11·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + (−0.858 − 0.512i)7-s − 0.817i·11-s + 1.81i·13-s − 0.371·17-s − 0.528i·19-s + 0.799i·23-s + 0.200·25-s − 0.711i·29-s − 0.584i·31-s + (−0.384 − 0.229i)35-s + 0.495·37-s + 1.02·41-s + 0.0713·43-s − 1.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0775 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0775 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.336498665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.336498665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (2.27 + 1.35i)T \) |
| good | 11 | \( 1 + 2.71iT - 11T^{2} \) |
| 13 | \( 1 - 6.54iT - 13T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 + 2.30iT - 19T^{2} \) |
| 23 | \( 1 - 3.83iT - 23T^{2} \) |
| 29 | \( 1 + 3.83iT - 29T^{2} \) |
| 31 | \( 1 + 3.25iT - 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 - 6.54T + 41T^{2} \) |
| 43 | \( 1 - 0.468T + 43T^{2} \) |
| 47 | \( 1 + 9.11T + 47T^{2} \) |
| 53 | \( 1 + 9.25iT - 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 4.78iT - 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 2.30iT - 71T^{2} \) |
| 73 | \( 1 + 11.4iT - 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 + 16.9iT - 97T^{2} \) |
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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
−8.032551821027752006811001095836, −7.24400159974971966035019544369, −6.43240544048408950993217652969, −6.22221222488612593559146041995, −5.12822031660883734133771011915, −4.24658542248441170412110446236, −3.60464186979851500010406214014, −2.62063241446679198416324133538, −1.69149496328581746890417087980, −0.39773966855068571037268739920,
1.01742229181796740197859712136, 2.34456875018414958130068702301, 2.92190308729432987868673731552, 3.83176455289052024303678886196, 4.89448442656839279684388964444, 5.54443555940773531666336906867, 6.21722589798280880456943788915, 6.88008529312901628518226446983, 7.72948626841628310467012447921, 8.395910805513737059614022879206
