L(s) = 1 | + 2-s − 1.41·3-s + 4-s − 4.24·5-s − 1.41·6-s + 8-s − 0.999·9-s − 4.24·10-s − 11-s − 1.41·12-s + 6·15-s + 16-s + 5.65·17-s − 0.999·18-s − 4.24·20-s − 22-s + 6·23-s − 1.41·24-s + 12.9·25-s + 5.65·27-s + 2·29-s + 6·30-s + 1.41·31-s + 32-s + 1.41·33-s + 5.65·34-s − 0.999·36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.816·3-s + 0.5·4-s − 1.89·5-s − 0.577·6-s + 0.353·8-s − 0.333·9-s − 1.34·10-s − 0.301·11-s − 0.408·12-s + 1.54·15-s + 0.250·16-s + 1.37·17-s − 0.235·18-s − 0.948·20-s − 0.213·22-s + 1.25·23-s − 0.288·24-s + 2.59·25-s + 1.08·27-s + 0.371·29-s + 1.09·30-s + 0.254·31-s + 0.176·32-s + 0.246·33-s + 0.970·34-s − 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.158654401\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.158654401\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−10.31712977353414038389342658987, −8.833869790839060640010182867948, −8.007950184997369309371357323402, −7.31380020293928955471865897045, −6.50040795053666263347425379635, −5.33161816403917439284993844242, −4.79434884703651495756367859786, −3.67085440498159200895188491665, −2.99064267300105637449759917029, −0.76692540560466746681841310049,
0.76692540560466746681841310049, 2.99064267300105637449759917029, 3.67085440498159200895188491665, 4.79434884703651495756367859786, 5.33161816403917439284993844242, 6.50040795053666263347425379635, 7.31380020293928955471865897045, 8.007950184997369309371357323402, 8.833869790839060640010182867948, 10.31712977353414038389342658987