| L(s) = 1 | + 1.41·2-s − 3-s − 5-s − 1.41·6-s − 0.828·7-s − 2.82·8-s + 9-s − 1.41·10-s − 2.41·11-s − 5.41·13-s − 1.17·14-s + 15-s − 4.00·16-s − 3.41·17-s + 1.41·18-s + 19-s + 0.828·21-s − 3.41·22-s + 2.82·24-s + 25-s − 7.65·26-s − 27-s + 0.828·29-s + 1.41·30-s − 3.82·31-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.577·3-s − 0.447·5-s − 0.577·6-s − 0.313·7-s − 0.999·8-s + 0.333·9-s − 0.447·10-s − 0.727·11-s − 1.50·13-s − 0.313·14-s + 0.258·15-s − 1.00·16-s − 0.828·17-s + 0.333·18-s + 0.229·19-s + 0.180·21-s − 0.727·22-s + 0.577·24-s + 0.200·25-s − 1.50·26-s − 0.192·27-s + 0.153·29-s + 0.258·30-s − 0.687·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5196119753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5196119753\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 + 3.41T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 0.343T + 59T^{2} \) |
| 61 | \( 1 - 1.34T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 4.89T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.69195576935134179751917693591, −6.90436476252576421025650024998, −6.42901175898127330446966260468, −5.46862111302589147013609864034, −4.97239409426428862037609211382, −4.54671964490362894953744010860, −3.61165222496310179045724319376, −2.93306224878706575590145107869, −2.02302001038406399086236850305, −0.29511055490017768506635106787,
0.29511055490017768506635106787, 2.02302001038406399086236850305, 2.93306224878706575590145107869, 3.61165222496310179045724319376, 4.54671964490362894953744010860, 4.97239409426428862037609211382, 5.46862111302589147013609864034, 6.42901175898127330446966260468, 6.90436476252576421025650024998, 7.69195576935134179751917693591
