L(s) = 1 | + 0.487·2-s + 0.235·3-s − 1.76·4-s + 5-s + 0.115·6-s − 1.09·7-s − 1.83·8-s − 2.94·9-s + 0.487·10-s − 1.06·11-s − 0.415·12-s + 3.57·13-s − 0.535·14-s + 0.235·15-s + 2.62·16-s − 17-s − 1.43·18-s − 0.200·19-s − 1.76·20-s − 0.258·21-s − 0.519·22-s + 2.04·23-s − 0.433·24-s + 25-s + 1.74·26-s − 1.40·27-s + 1.93·28-s + ⋯ |
L(s) = 1 | + 0.345·2-s + 0.136·3-s − 0.880·4-s + 0.447·5-s + 0.0470·6-s − 0.414·7-s − 0.649·8-s − 0.981·9-s + 0.154·10-s − 0.320·11-s − 0.119·12-s + 0.990·13-s − 0.142·14-s + 0.0609·15-s + 0.656·16-s − 0.242·17-s − 0.338·18-s − 0.0460·19-s − 0.393·20-s − 0.0564·21-s − 0.110·22-s + 0.426·23-s − 0.0884·24-s + 0.200·25-s + 0.341·26-s − 0.269·27-s + 0.365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
good | 2 | \( 1 - 0.487T + 2T^{2} \) |
| 3 | \( 1 - 0.235T + 3T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 11 | \( 1 + 1.06T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 19 | \( 1 + 0.200T + 19T^{2} \) |
| 23 | \( 1 - 2.04T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 - 8.58T + 31T^{2} \) |
| 37 | \( 1 + 0.400T + 37T^{2} \) |
| 41 | \( 1 + 4.45T + 41T^{2} \) |
| 43 | \( 1 - 1.47T + 43T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 + 0.707T + 53T^{2} \) |
| 59 | \( 1 - 4.46T + 59T^{2} \) |
| 61 | \( 1 + 4.33T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 73 | \( 1 + 2.93T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 5.73T + 89T^{2} \) |
| 97 | \( 1 - 2.53T + 97T^{2} \) |
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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
−8.042089860983930503543111714612, −6.75217327421421267887622706126, −6.20704024187498287314432027116, −5.52891788614439656713044585174, −4.90446637522889569709231748042, −4.03449343615039788762235734603, −3.20308509126918151880261521698, −2.63849388389185261618826224654, −1.22826114953618817127554434832, 0,
1.22826114953618817127554434832, 2.63849388389185261618826224654, 3.20308509126918151880261521698, 4.03449343615039788762235734603, 4.90446637522889569709231748042, 5.52891788614439656713044585174, 6.20704024187498287314432027116, 6.75217327421421267887622706126, 8.042089860983930503543111714612