L(s) = 1 | + 1.67·2-s + 3-s + 0.800·4-s + 1.67·6-s − 2.75·7-s − 2.00·8-s + 9-s − 3.09·11-s + 0.800·12-s + 4.10·13-s − 4.60·14-s − 4.96·16-s + 4.07·17-s + 1.67·18-s + 5.75·19-s − 2.75·21-s − 5.17·22-s + 6.80·23-s − 2.00·24-s + 6.86·26-s + 27-s − 2.20·28-s + 1.81·29-s + 3.24·31-s − 4.28·32-s − 3.09·33-s + 6.81·34-s + ⋯ |
L(s) = 1 | + 1.18·2-s + 0.577·3-s + 0.400·4-s + 0.683·6-s − 1.04·7-s − 0.709·8-s + 0.333·9-s − 0.932·11-s + 0.230·12-s + 1.13·13-s − 1.23·14-s − 1.24·16-s + 0.987·17-s + 0.394·18-s + 1.32·19-s − 0.600·21-s − 1.10·22-s + 1.41·23-s − 0.409·24-s + 1.34·26-s + 0.192·27-s − 0.416·28-s + 0.336·29-s + 0.583·31-s − 0.757·32-s − 0.538·33-s + 1.16·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.656708310\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.656708310\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 7 | \( 1 + 2.75T + 7T^{2} \) |
| 11 | \( 1 + 3.09T + 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 - 4.07T + 17T^{2} \) |
| 19 | \( 1 - 5.75T + 19T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 - 1.81T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 + 5.48T + 37T^{2} \) |
| 41 | \( 1 - 1.31T + 41T^{2} \) |
| 43 | \( 1 - 8.65T + 43T^{2} \) |
| 53 | \( 1 + 5.58T + 53T^{2} \) |
| 59 | \( 1 - 0.623T + 59T^{2} \) |
| 61 | \( 1 - 6.04T + 61T^{2} \) |
| 67 | \( 1 + 4.05T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 3.81T + 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 - 4.45T + 83T^{2} \) |
| 89 | \( 1 + 2.86T + 89T^{2} \) |
| 97 | \( 1 - 6.01T + 97T^{2} \) |
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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
−8.610581050864647670007035676271, −7.74019358158130414552024611754, −6.94172656652423324891925303979, −6.15183238237667179972005728181, −5.45191520736683810932520090605, −4.79679751195766746913479253773, −3.63056379508956768197619529738, −3.27553537257682733607828016382, −2.60026505834606179362179154168, −0.935460296389038525589942281066,
0.935460296389038525589942281066, 2.60026505834606179362179154168, 3.27553537257682733607828016382, 3.63056379508956768197619529738, 4.79679751195766746913479253773, 5.45191520736683810932520090605, 6.15183238237667179972005728181, 6.94172656652423324891925303979, 7.74019358158130414552024611754, 8.610581050864647670007035676271