L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 4.80·7-s − 8-s + 9-s + 10-s + 2.55·11-s + 12-s − 4.80·14-s − 15-s + 16-s − 3.02·17-s − 18-s − 3.89·19-s − 20-s + 4.80·21-s − 2.55·22-s − 1.64·23-s − 24-s + 25-s + 27-s + 4.80·28-s + 5.07·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.81·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.770·11-s + 0.288·12-s − 1.28·14-s − 0.258·15-s + 0.250·16-s − 0.734·17-s − 0.235·18-s − 0.894·19-s − 0.223·20-s + 1.04·21-s − 0.544·22-s − 0.342·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.907·28-s + 0.942·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.159290914\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.159290914\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 4.80T + 7T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 17 | \( 1 + 3.02T + 17T^{2} \) |
| 19 | \( 1 + 3.89T + 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 - 7.51T + 37T^{2} \) |
| 41 | \( 1 + 5.89T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 6.42T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 - 7.60T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 0.929T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 5.40T + 79T^{2} \) |
| 83 | \( 1 - 4.33T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−8.381086587859211177207195728916, −7.85889169676724704710059292166, −6.90736348814798056870876331202, −6.40776626153954386104477563313, −5.14432823229774307438050967178, −4.45761261521035848546731509054, −3.81135494100607934206314630857, −2.53418287910930099257937495119, −1.84579632741894035076039638235, −0.912579351530342337874080420322,
0.912579351530342337874080420322, 1.84579632741894035076039638235, 2.53418287910930099257937495119, 3.81135494100607934206314630857, 4.45761261521035848546731509054, 5.14432823229774307438050967178, 6.40776626153954386104477563313, 6.90736348814798056870876331202, 7.85889169676724704710059292166, 8.381086587859211177207195728916