L(s) = 1 | − 2·3-s + 4·5-s + 7-s + 9-s + 4·13-s − 8·15-s + 17-s − 6·19-s − 2·21-s + 11·25-s + 4·27-s − 6·29-s − 4·31-s + 4·35-s − 10·37-s − 8·39-s − 6·41-s + 4·45-s − 4·47-s + 49-s − 2·51-s + 14·53-s + 12·57-s + 6·59-s + 12·61-s + 63-s + 16·65-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 2.06·15-s + 0.242·17-s − 1.37·19-s − 0.436·21-s + 11/5·25-s + 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.676·35-s − 1.64·37-s − 1.28·39-s − 0.937·41-s + 0.596·45-s − 0.583·47-s + 1/7·49-s − 0.280·51-s + 1.92·53-s + 1.58·57-s + 0.781·59-s + 1.53·61-s + 0.125·63-s + 1.98·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.11091127878517, −12.77520062108109, −12.28203068784104, −11.60689451831180, −11.32191541323329, −10.71687494627129, −10.40283948878485, −10.19383594229154, −9.464055268139615, −8.889445596206219, −8.647632802559292, −8.131092827111619, −7.184738243927582, −6.750879880913125, −6.417679626686825, −5.846731931215474, −5.539239739703842, −5.189941600491000, −4.679215940210337, −3.802518213795480, −3.449070569346722, −2.382480218761390, −2.059507492760715, −1.478111438716098, −0.8775591341690060, 0,
0.8775591341690060, 1.478111438716098, 2.059507492760715, 2.382480218761390, 3.449070569346722, 3.802518213795480, 4.679215940210337, 5.189941600491000, 5.539239739703842, 5.846731931215474, 6.417679626686825, 6.750879880913125, 7.184738243927582, 8.131092827111619, 8.647632802559292, 8.889445596206219, 9.464055268139615, 10.19383594229154, 10.40283948878485, 10.71687494627129, 11.32191541323329, 11.60689451831180, 12.28203068784104, 12.77520062108109, 13.11091127878517