| L(s) = 1 | − 0.504·2-s − 7.74·4-s − 7·7-s + 7.94·8-s − 54.8·11-s − 16.0·13-s + 3.53·14-s + 57.9·16-s − 0.422·17-s + 127.·19-s + 27.7·22-s + 51.1·23-s + 8.08·26-s + 54.2·28-s − 41.4·29-s + 192.·31-s − 92.8·32-s + 0.213·34-s + 189.·37-s − 64.3·38-s + 76.3·41-s − 294.·43-s + 425.·44-s − 25.7·46-s + 540.·47-s + 49·49-s + 123.·52-s + ⋯ |
| L(s) = 1 | − 0.178·2-s − 0.968·4-s − 0.377·7-s + 0.351·8-s − 1.50·11-s − 0.341·13-s + 0.0674·14-s + 0.905·16-s − 0.00602·17-s + 1.53·19-s + 0.268·22-s + 0.463·23-s + 0.0609·26-s + 0.365·28-s − 0.265·29-s + 1.11·31-s − 0.512·32-s + 0.00107·34-s + 0.840·37-s − 0.274·38-s + 0.290·41-s − 1.04·43-s + 1.45·44-s − 0.0826·46-s + 1.67·47-s + 0.142·49-s + 0.330·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 0.504T + 8T^{2} \) |
| 11 | \( 1 + 54.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 0.422T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 51.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 192.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 76.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 294.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 540.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 661.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 410.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 46.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 10.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 491.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 814.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 858.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 341.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−8.692838090400111839229825271680, −7.83725063916199787511581237933, −7.37465796347006608913571561343, −6.07130404046609051719066019444, −5.21095000228852869295322055771, −4.66043961227287916805022355410, −3.43658528816878639627630275759, −2.62286434407456940370808181808, −1.03205557113987557879408677893, 0,
1.03205557113987557879408677893, 2.62286434407456940370808181808, 3.43658528816878639627630275759, 4.66043961227287916805022355410, 5.21095000228852869295322055771, 6.07130404046609051719066019444, 7.37465796347006608913571561343, 7.83725063916199787511581237933, 8.692838090400111839229825271680
