L(s) = 1 | − 2-s − 1.90·3-s + 4-s + 3.90·5-s + 1.90·6-s + 1.35·7-s − 8-s + 0.630·9-s − 3.90·10-s + 4.12·11-s − 1.90·12-s + 0.627·13-s − 1.35·14-s − 7.44·15-s + 16-s + 5.89·17-s − 0.630·18-s + 6.12·19-s + 3.90·20-s − 2.57·21-s − 4.12·22-s + 23-s + 1.90·24-s + 10.2·25-s − 0.627·26-s + 4.51·27-s + 1.35·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.10·3-s + 0.5·4-s + 1.74·5-s + 0.777·6-s + 0.510·7-s − 0.353·8-s + 0.210·9-s − 1.23·10-s + 1.24·11-s − 0.550·12-s + 0.174·13-s − 0.361·14-s − 1.92·15-s + 0.250·16-s + 1.42·17-s − 0.148·18-s + 1.40·19-s + 0.874·20-s − 0.561·21-s − 0.879·22-s + 0.208·23-s + 0.388·24-s + 2.05·25-s − 0.123·26-s + 0.868·27-s + 0.255·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.928437604\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.928437604\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 1.90T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 - 4.12T + 11T^{2} \) |
| 13 | \( 1 - 0.627T + 13T^{2} \) |
| 17 | \( 1 - 5.89T + 17T^{2} \) |
| 19 | \( 1 - 6.12T + 19T^{2} \) |
| 29 | \( 1 + 2.99T + 29T^{2} \) |
| 31 | \( 1 + 0.570T + 31T^{2} \) |
| 37 | \( 1 - 5.63T + 37T^{2} \) |
| 41 | \( 1 + 0.123T + 41T^{2} \) |
| 43 | \( 1 - 9.95T + 43T^{2} \) |
| 47 | \( 1 + 2.45T + 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 - 0.881T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 + 9.11T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 73 | \( 1 + 0.290T + 73T^{2} \) |
| 79 | \( 1 - 4.59T + 79T^{2} \) |
| 83 | \( 1 + 1.08T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.073334025417495350932411027567, −7.26617693428325563686499849860, −6.49921729576350466526711215980, −5.91933749749985997420363635266, −5.51700749158820391317370066590, −4.83585338711205399447115241599, −3.51020111296865506561910963377, −2.50389321020783001356289440870, −1.34824264193060318123650872936, −1.05823476267456939908517285336,
1.05823476267456939908517285336, 1.34824264193060318123650872936, 2.50389321020783001356289440870, 3.51020111296865506561910963377, 4.83585338711205399447115241599, 5.51700749158820391317370066590, 5.91933749749985997420363635266, 6.49921729576350466526711215980, 7.26617693428325563686499849860, 8.073334025417495350932411027567