| L(s) = 1 | + 102·3-s + 6.18e3·7-s + 3.84e3·9-s − 1.45e4·13-s + 1.60e5·19-s + 6.31e5·21-s − 5.01e5·25-s − 2.77e5·27-s − 8.71e5·31-s + 2.31e6·37-s − 1.48e6·39-s − 1.98e6·43-s + 1.71e7·49-s + 1.63e7·57-s + 3.87e7·61-s + 2.37e7·63-s + 5.60e7·67-s − 5.04e7·73-s − 5.11e7·75-s + 1.26e8·79-s − 5.34e7·81-s − 9.02e7·91-s − 8.89e7·93-s + 3.91e7·97-s − 2.26e8·103-s + 3.48e6·109-s + 2.36e8·111-s + ⋯ |
| L(s) = 1 | + 1.25·3-s + 2.57·7-s + 0.585·9-s − 0.510·13-s + 1.23·19-s + 3.24·21-s − 1.28·25-s − 0.521·27-s − 0.944·31-s + 1.23·37-s − 0.643·39-s − 0.579·43-s + 2.98·49-s + 1.55·57-s + 2.79·61-s + 1.50·63-s + 2.78·67-s − 1.77·73-s − 1.61·75-s + 3.25·79-s − 1.24·81-s − 1.31·91-s − 1.18·93-s + 0.441·97-s − 2.01·103-s + 0.0247·109-s + 1.55·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(6.165356975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.165356975\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 34 p T + p^{8} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 100358 p T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 442 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 38857702 p T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 7294 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 10482174722 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 80326 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 147132606722 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 253546712162 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 435914 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1159298 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8591852110082 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 990266 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2717384513282 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 23095221504482 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 291106443928802 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 19369154 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 28024294 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 158683081351682 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 25230142 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 63401398 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2244210903661922 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1743003813196802 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19550306 T + p^{8} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.58730706292504581073874212453, −13.75288103228205565243398398098, −13.40195829448320810875468501270, −12.47796383522339942975958291998, −11.56513744443692035526316497590, −11.50055096593137212761747661019, −10.81283630298448919769436886886, −9.734876017079594824441781513070, −9.459384491369489843571844499675, −8.417552930180819718903983809713, −8.140818590203939019187592822439, −7.69840696234870427058244988647, −7.04607653732385909645602265984, −5.58991514967590570170890589196, −5.10493624139923443195004695612, −4.26688460679142598705277960450, −3.46511132240332600558733629769, −2.24852024528193570998834094806, −1.88561001509159264518862070429, −0.880986730195474693320427250705,
0.880986730195474693320427250705, 1.88561001509159264518862070429, 2.24852024528193570998834094806, 3.46511132240332600558733629769, 4.26688460679142598705277960450, 5.10493624139923443195004695612, 5.58991514967590570170890589196, 7.04607653732385909645602265984, 7.69840696234870427058244988647, 8.140818590203939019187592822439, 8.417552930180819718903983809713, 9.459384491369489843571844499675, 9.734876017079594824441781513070, 10.81283630298448919769436886886, 11.50055096593137212761747661019, 11.56513744443692035526316497590, 12.47796383522339942975958291998, 13.40195829448320810875468501270, 13.75288103228205565243398398098, 14.58730706292504581073874212453
