| L(s) = 1 | − 0.347·2-s − 3-s − 1.87·4-s + 0.652·5-s + 0.347·6-s + 1.34·8-s + 9-s − 0.226·10-s + 0.532·11-s + 1.87·12-s + 13-s − 0.652·15-s + 3.29·16-s + 1.12·17-s − 0.347·18-s + 0.305·19-s − 1.22·20-s − 0.184·22-s − 8.00·23-s − 1.34·24-s − 4.57·25-s − 0.347·26-s − 27-s + 7.78·29-s + 0.226·30-s + 0.588·31-s − 3.83·32-s + ⋯ |
| L(s) = 1 | − 0.245·2-s − 0.577·3-s − 0.939·4-s + 0.291·5-s + 0.141·6-s + 0.476·8-s + 0.333·9-s − 0.0716·10-s + 0.160·11-s + 0.542·12-s + 0.277·13-s − 0.168·15-s + 0.822·16-s + 0.271·17-s − 0.0818·18-s + 0.0700·19-s − 0.274·20-s − 0.0393·22-s − 1.66·23-s − 0.275·24-s − 0.914·25-s − 0.0681·26-s − 0.192·27-s + 1.44·29-s + 0.0413·30-s + 0.105·31-s − 0.678·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9397138691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9397138691\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.347T + 2T^{2} \) |
| 5 | \( 1 - 0.652T + 5T^{2} \) |
| 11 | \( 1 - 0.532T + 11T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 - 0.305T + 19T^{2} \) |
| 23 | \( 1 + 8.00T + 23T^{2} \) |
| 29 | \( 1 - 7.78T + 29T^{2} \) |
| 31 | \( 1 - 0.588T + 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 - 2.94T + 43T^{2} \) |
| 47 | \( 1 + 4.12T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 - 4.66T + 67T^{2} \) |
| 71 | \( 1 + 3.92T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 2.46T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−9.320190995120401635628178989538, −8.371296387715298238835245500631, −7.87784625822147612283131116403, −6.73857845316536487488316100662, −5.93342925166333747995307426324, −5.22028221168607774259301095043, −4.32083004051057886733360121592, −3.55886805413693715940757089292, −1.98645625693714801795880744869, −0.71232380683229032364762490047,
0.71232380683229032364762490047, 1.98645625693714801795880744869, 3.55886805413693715940757089292, 4.32083004051057886733360121592, 5.22028221168607774259301095043, 5.93342925166333747995307426324, 6.73857845316536487488316100662, 7.87784625822147612283131116403, 8.371296387715298238835245500631, 9.320190995120401635628178989538
