L(s) = 1 | + 1.41i·2-s − 1.00·4-s + 1.41i·7-s + 11-s + 1.41i·13-s − 2.00·14-s − 0.999·16-s − 1.41i·17-s + 1.41i·22-s − 2.00·26-s − 1.41i·28-s − 1.41i·32-s + 2.00·34-s + 1.41i·43-s − 1.00·44-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 1.00·4-s + 1.41i·7-s + 11-s + 1.41i·13-s − 2.00·14-s − 0.999·16-s − 1.41i·17-s + 1.41i·22-s − 2.00·26-s − 1.41i·28-s − 1.41i·32-s + 2.00·34-s + 1.41i·43-s − 1.00·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.184399837\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.184399837\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 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
−9.148943117436535000733471307183, −8.793860599828825712313843901958, −7.87279740722943014216247263140, −7.01917485879440151615211308252, −6.47194816248568494855302905191, −5.82854231331736859325744871499, −4.98190229141573149935883669302, −4.31087894259796945565522230331, −2.89133282975621820332282598464, −1.85598143194806943782795038501,
0.830067582061538016029864266451, 1.70428658917981024818608436446, 2.98543465640059473653356308600, 3.86423390662753462415709507789, 4.14989250529904364092095819238, 5.43950039264207885793310037888, 6.50316908836612358804794293084, 7.21442510577589526809801131629, 8.139626582578233471673799587764, 8.897213897461852314347870728101