| L(s) = 1 | − 2-s + 1.68·3-s + 4-s + 4.36·5-s − 1.68·6-s − 8-s − 0.150·9-s − 4.36·10-s − 0.568·11-s + 1.68·12-s − 3.23·13-s + 7.36·15-s + 16-s + 7.22·17-s + 0.150·18-s + 3.92·19-s + 4.36·20-s + 0.568·22-s + 0.128·23-s − 1.68·24-s + 14.0·25-s + 3.23·26-s − 5.31·27-s + 0.183·29-s − 7.36·30-s − 0.398·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.974·3-s + 0.5·4-s + 1.95·5-s − 0.689·6-s − 0.353·8-s − 0.0502·9-s − 1.37·10-s − 0.171·11-s + 0.487·12-s − 0.897·13-s + 1.90·15-s + 0.250·16-s + 1.75·17-s + 0.0355·18-s + 0.901·19-s + 0.975·20-s + 0.121·22-s + 0.0267·23-s − 0.344·24-s + 2.80·25-s + 0.634·26-s − 1.02·27-s + 0.0340·29-s − 1.34·30-s − 0.0715·31-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
| \(L(1)\) |
\(\approx\) |
\(2.934926547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.934926547\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.68T + 3T^{2} \) |
| 5 | \( 1 - 4.36T + 5T^{2} \) |
| 11 | \( 1 + 0.568T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 7.22T + 17T^{2} \) |
| 19 | \( 1 - 3.92T + 19T^{2} \) |
| 23 | \( 1 - 0.128T + 23T^{2} \) |
| 29 | \( 1 - 0.183T + 29T^{2} \) |
| 31 | \( 1 + 0.398T + 31T^{2} \) |
| 37 | \( 1 - 1.28T + 37T^{2} \) |
| 43 | \( 1 - 6.36T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 + 4.31T + 53T^{2} \) |
| 59 | \( 1 + 0.102T + 59T^{2} \) |
| 61 | \( 1 + 4.87T + 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 - 1.98T + 83T^{2} \) |
| 89 | \( 1 + 5.16T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.618124049401479721426016393914, −7.74298509335141293774821568420, −7.27196871850847552989821606057, −6.16544039296486285081663075936, −5.63769174883293100885528132598, −4.93903494097329224531594325300, −3.32726343379447021917192128100, −2.73722433337050528181913639691, −2.02366365906532723513171419181, −1.10860262275933176915466696128,
1.10860262275933176915466696128, 2.02366365906532723513171419181, 2.73722433337050528181913639691, 3.32726343379447021917192128100, 4.93903494097329224531594325300, 5.63769174883293100885528132598, 6.16544039296486285081663075936, 7.27196871850847552989821606057, 7.74298509335141293774821568420, 8.618124049401479721426016393914
