L(s) = 1 | − 1.41i·3-s + 5-s + 7-s − 1.00·9-s − 11-s − 1.41i·15-s + 17-s − 19-s − 1.41i·21-s − 1.41i·29-s − 1.41i·31-s + 1.41i·33-s + 35-s + 1.41i·37-s + 1.41i·41-s + ⋯ |
L(s) = 1 | − 1.41i·3-s + 5-s + 7-s − 1.00·9-s − 11-s − 1.41i·15-s + 17-s − 19-s − 1.41i·21-s − 1.41i·29-s − 1.41i·31-s + 1.41i·33-s + 35-s + 1.41i·37-s + 1.41i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.510196287\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.510196287\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - 1.41iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.653094923891008206358254448164, −7.87590664909769522954618263760, −7.70912239580716861992538208203, −6.53998668565952309128283896751, −5.94765366757711923672069878061, −5.27815568994135145061491950944, −4.23152317681079038390802354369, −2.59245854390906704341164083930, −2.10761664994268906131972631424, −1.10537673504947403609078840838,
1.68289640088222024848000251785, 2.73247974143644809381584538023, 3.79306020109889461278852983434, 4.67957744370422299727739078929, 5.39273852645886723190454549553, 5.73688091571638148801460200408, 7.09110441015822124429419541493, 7.938191712881340308333911530042, 8.872935448251216144919976606266, 9.250953202723730190323717888269