L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 2·10-s + 2·11-s − 4·13-s + 14-s + 16-s + 6·17-s + 2·20-s − 2·22-s − 23-s − 25-s + 4·26-s − 28-s + 2·29-s + 4·31-s − 32-s − 6·34-s − 2·35-s − 2·40-s − 6·41-s + 6·43-s + 2·44-s + 46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s + 0.603·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.447·20-s − 0.426·22-s − 0.208·23-s − 1/5·25-s + 0.784·26-s − 0.188·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.338·35-s − 0.316·40-s − 0.937·41-s + 0.914·43-s + 0.301·44-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.517943935\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.517943935\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.920992901098182601901086017973, −8.036211124547257201070037966541, −7.30195511007121109487420606828, −6.56834029106615434473661999862, −5.81611514957424867826659470627, −5.12076469021786144835353560893, −3.88707904313419876315262513699, −2.84617567899143688107199956239, −1.99547598448117038089284427977, −0.854615802807522975295731889880,
0.854615802807522975295731889880, 1.99547598448117038089284427977, 2.84617567899143688107199956239, 3.88707904313419876315262513699, 5.12076469021786144835353560893, 5.81611514957424867826659470627, 6.56834029106615434473661999862, 7.30195511007121109487420606828, 8.036211124547257201070037966541, 8.920992901098182601901086017973