L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 6.24·11-s − 12-s − 5.65·13-s + 16-s + 5.41·17-s + 18-s + 1.17·19-s − 6.24·22-s − 8.82·23-s − 24-s − 5.65·26-s − 27-s + 5.41·29-s + 1.75·31-s + 32-s + 6.24·33-s + 5.41·34-s + 36-s − 8.24·37-s + 1.17·38-s + 5.65·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.88·11-s − 0.288·12-s − 1.56·13-s + 0.250·16-s + 1.31·17-s + 0.235·18-s + 0.268·19-s − 1.33·22-s − 1.84·23-s − 0.204·24-s − 1.10·26-s − 0.192·27-s + 1.00·29-s + 0.315·31-s + 0.176·32-s + 1.08·33-s + 0.928·34-s + 0.166·36-s − 1.35·37-s + 0.190·38-s + 0.905·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.746335382\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.746335382\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 - 5.07T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.74891775850857897164559316990, −7.27937661176114399878793193550, −6.33091701021095749697933958644, −5.58249762861459552166960893998, −5.16544639333246706434988218372, −4.58041051025677143173928819918, −3.59119478652307199329994748015, −2.67680219011026840865841170826, −2.09152759847903227016656292491, −0.58336123294360954269633659691,
0.58336123294360954269633659691, 2.09152759847903227016656292491, 2.67680219011026840865841170826, 3.59119478652307199329994748015, 4.58041051025677143173928819918, 5.16544639333246706434988218372, 5.58249762861459552166960893998, 6.33091701021095749697933958644, 7.27937661176114399878793193550, 7.74891775850857897164559316990