L(s) = 1 | − 3.08·3-s − 5-s − 1.91·7-s + 6.49·9-s − 1.79·11-s − 0.701·13-s + 3.08·15-s + 0.463·17-s + 0.120·19-s + 5.90·21-s − 0.408·23-s + 25-s − 10.7·27-s + 1.24·29-s + 2.54·31-s + 5.53·33-s + 1.91·35-s − 37-s + 2.16·39-s + 9.54·41-s + 4.56·43-s − 6.49·45-s + 7.05·47-s − 3.32·49-s − 1.42·51-s − 3.77·53-s + 1.79·55-s + ⋯ |
L(s) = 1 | − 1.77·3-s − 0.447·5-s − 0.724·7-s + 2.16·9-s − 0.541·11-s − 0.194·13-s + 0.795·15-s + 0.112·17-s + 0.0276·19-s + 1.28·21-s − 0.0852·23-s + 0.200·25-s − 2.07·27-s + 0.230·29-s + 0.457·31-s + 0.963·33-s + 0.323·35-s − 0.164·37-s + 0.346·39-s + 1.49·41-s + 0.695·43-s − 0.968·45-s + 1.02·47-s − 0.475·49-s − 0.200·51-s − 0.517·53-s + 0.242·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 7 | \( 1 + 1.91T + 7T^{2} \) |
| 11 | \( 1 + 1.79T + 11T^{2} \) |
| 13 | \( 1 + 0.701T + 13T^{2} \) |
| 17 | \( 1 - 0.463T + 17T^{2} \) |
| 19 | \( 1 - 0.120T + 19T^{2} \) |
| 23 | \( 1 + 0.408T + 23T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 41 | \( 1 - 9.54T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 47 | \( 1 - 7.05T + 47T^{2} \) |
| 53 | \( 1 + 3.77T + 53T^{2} \) |
| 59 | \( 1 + 1.54T + 59T^{2} \) |
| 61 | \( 1 - 8.89T + 61T^{2} \) |
| 67 | \( 1 + 3.19T + 67T^{2} \) |
| 71 | \( 1 - 4.19T + 71T^{2} \) |
| 73 | \( 1 - 6.19T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.195843894467912411693494901354, −7.38140520119390447514463922463, −6.73259521533354047163298711549, −6.03309227255381808896958874976, −5.39505960282335512821869946578, −4.60622441515514835255854847045, −3.82822632350122399359903652440, −2.58805966324303704700565235936, −1.01536208037176210836416163358, 0,
1.01536208037176210836416163358, 2.58805966324303704700565235936, 3.82822632350122399359903652440, 4.60622441515514835255854847045, 5.39505960282335512821869946578, 6.03309227255381808896958874976, 6.73259521533354047163298711549, 7.38140520119390447514463922463, 8.195843894467912411693494901354