| L(s) = 1 | − 0.593·3-s + 2.48·5-s − 3.58·7-s − 2.64·9-s + 0.309·11-s − 1.08·13-s − 1.47·15-s − 5.71·17-s + 1.82·19-s + 2.12·21-s + 1.16·25-s + 3.35·27-s − 6.00·29-s + 6.94·31-s − 0.183·33-s − 8.89·35-s + 9.89·37-s + 0.640·39-s − 6.59·41-s + 2.87·43-s − 6.57·45-s + 12.3·47-s + 5.84·49-s + 3.39·51-s − 8.28·53-s + 0.768·55-s − 1.08·57-s + ⋯ |
| L(s) = 1 | − 0.342·3-s + 1.11·5-s − 1.35·7-s − 0.882·9-s + 0.0933·11-s − 0.299·13-s − 0.380·15-s − 1.38·17-s + 0.418·19-s + 0.463·21-s + 0.232·25-s + 0.644·27-s − 1.11·29-s + 1.24·31-s − 0.0319·33-s − 1.50·35-s + 1.62·37-s + 0.102·39-s − 1.03·41-s + 0.438·43-s − 0.979·45-s + 1.80·47-s + 0.834·49-s + 0.475·51-s − 1.13·53-s + 0.103·55-s − 0.143·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.240879667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.240879667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.593T + 3T^{2} \) |
| 5 | \( 1 - 2.48T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 - 0.309T + 11T^{2} \) |
| 13 | \( 1 + 1.08T + 13T^{2} \) |
| 17 | \( 1 + 5.71T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 29 | \( 1 + 6.00T + 29T^{2} \) |
| 31 | \( 1 - 6.94T + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 + 6.59T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 8.28T + 53T^{2} \) |
| 59 | \( 1 - 3.48T + 59T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 + 1.48T + 67T^{2} \) |
| 71 | \( 1 - 5.32T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 3.51T + 83T^{2} \) |
| 89 | \( 1 + 1.08T + 89T^{2} \) |
| 97 | \( 1 + 1.05T + 97T^{2} \) |
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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
−8.614029008194373130451788952174, −7.56015229930667886770431518894, −6.60652388712892598192833019387, −6.23418197269000551888221444487, −5.64774371813151023731002896318, −4.80411140848860628322282434899, −3.73921340284468086737827494923, −2.75333524308582244118402865350, −2.17667177092094854434426874719, −0.60853710882646040868322592327,
0.60853710882646040868322592327, 2.17667177092094854434426874719, 2.75333524308582244118402865350, 3.73921340284468086737827494923, 4.80411140848860628322282434899, 5.64774371813151023731002896318, 6.23418197269000551888221444487, 6.60652388712892598192833019387, 7.56015229930667886770431518894, 8.614029008194373130451788952174
