L(s) = 1 | + 2·2-s + 2·4-s + 5-s − 7-s − 3·9-s + 2·10-s − 2·11-s − 2·13-s − 2·14-s − 4·16-s − 3·17-s − 6·18-s + 2·19-s + 2·20-s − 4·22-s + 25-s − 4·26-s − 2·28-s + 7·29-s − 5·31-s − 8·32-s − 6·34-s − 35-s − 6·36-s − 11·37-s + 4·38-s + 41-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s − 9-s + 0.632·10-s − 0.603·11-s − 0.554·13-s − 0.534·14-s − 16-s − 0.727·17-s − 1.41·18-s + 0.458·19-s + 0.447·20-s − 0.852·22-s + 1/5·25-s − 0.784·26-s − 0.377·28-s + 1.29·29-s − 0.898·31-s − 1.41·32-s − 1.02·34-s − 0.169·35-s − 36-s − 1.80·37-s + 0.648·38-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.551506056517706907038082013704, −7.49497922965664040486613733245, −6.56389614092243760139893628224, −6.05404124546789425257449477618, −5.13478410924635941128044754757, −4.81791042909742149414815821233, −3.52621673706726526297889832836, −2.92211424763431944963205262512, −2.08330470038331107592069998888, 0,
2.08330470038331107592069998888, 2.92211424763431944963205262512, 3.52621673706726526297889832836, 4.81791042909742149414815821233, 5.13478410924635941128044754757, 6.05404124546789425257449477618, 6.56389614092243760139893628224, 7.49497922965664040486613733245, 8.551506056517706907038082013704