L(s) = 1 | − 2-s − 2.84·3-s + 4-s − 0.100·5-s + 2.84·6-s − 8-s + 5.09·9-s + 0.100·10-s + 5.49·11-s − 2.84·12-s − 2.00·13-s + 0.284·15-s + 16-s + 2.02·17-s − 5.09·18-s − 5.25·19-s − 0.100·20-s − 5.49·22-s − 0.177·23-s + 2.84·24-s − 4.98·25-s + 2.00·26-s − 5.95·27-s + 1.92·29-s − 0.284·30-s + 1.74·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.64·3-s + 0.5·4-s − 0.0447·5-s + 1.16·6-s − 0.353·8-s + 1.69·9-s + 0.0316·10-s + 1.65·11-s − 0.821·12-s − 0.555·13-s + 0.0734·15-s + 0.250·16-s + 0.491·17-s − 1.20·18-s − 1.20·19-s − 0.0223·20-s − 1.17·22-s − 0.0370·23-s + 0.580·24-s − 0.997·25-s + 0.393·26-s − 1.14·27-s + 0.357·29-s − 0.0519·30-s + 0.313·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 2.84T + 3T^{2} \) |
| 5 | \( 1 + 0.100T + 5T^{2} \) |
| 11 | \( 1 - 5.49T + 11T^{2} \) |
| 13 | \( 1 + 2.00T + 13T^{2} \) |
| 17 | \( 1 - 2.02T + 17T^{2} \) |
| 19 | \( 1 + 5.25T + 19T^{2} \) |
| 23 | \( 1 + 0.177T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 - 1.74T + 31T^{2} \) |
| 37 | \( 1 - 0.296T + 37T^{2} \) |
| 43 | \( 1 + 3.48T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 0.522T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 7.52T + 67T^{2} \) |
| 71 | \( 1 - 0.0873T + 71T^{2} \) |
| 73 | \( 1 - 7.69T + 73T^{2} \) |
| 79 | \( 1 + 2.15T + 79T^{2} \) |
| 83 | \( 1 - 1.63T + 83T^{2} \) |
| 89 | \( 1 - 6.70T + 89T^{2} \) |
| 97 | \( 1 - 0.783T + 97T^{2} \) |
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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
−8.060038806762500002248753480222, −7.15438500603958449465140855783, −6.43163446199272307921708974904, −6.21712155844546703750839614785, −5.19400612619002031058546903393, −4.42439909355609736176287029868, −3.56216700288163890091040205254, −2.01655064910777478848848104919, −1.10468887063430229067302196370, 0,
1.10468887063430229067302196370, 2.01655064910777478848848104919, 3.56216700288163890091040205254, 4.42439909355609736176287029868, 5.19400612619002031058546903393, 6.21712155844546703750839614785, 6.43163446199272307921708974904, 7.15438500603958449465140855783, 8.060038806762500002248753480222