| L(s) = 1 | + 3-s + 4-s + 7-s + 12-s + 3·13-s + 21-s − 27-s + 28-s − 2·37-s + 3·39-s + 49-s + 3·52-s − 64-s + 67-s − 2·73-s − 3·79-s − 81-s + 84-s + 3·91-s − 108-s − 3·109-s − 2·111-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s + 7-s + 12-s + 3·13-s + 21-s − 27-s + 28-s − 2·37-s + 3·39-s + 49-s + 3·52-s − 64-s + 67-s − 2·73-s − 3·79-s − 81-s + 84-s + 3·91-s − 108-s − 3·109-s − 2·111-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7700625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7700625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.156772993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.156772993\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−8.852073757962957248510621342561, −8.737746830402323761778979237602, −8.558061667358138619338182477038, −7.978756701008942807421859585060, −7.971484709414468762173790282772, −7.12322501217349445116362600144, −7.11804143952243880207139070366, −6.60572401394442785051130352317, −5.98000098094200671993789129484, −5.92538701598272688267222747163, −5.44634112571062042917450517953, −4.92486245114663513451730362960, −4.24079695779039540127202302746, −3.91341765843496044833779666399, −3.59762279093051355611247593360, −2.94763947132601989060785544738, −2.78150680868566652829432988207, −1.91908160474091000860488401732, −1.64491124363510222152541210383, −1.23657015470512178342727386420,
1.23657015470512178342727386420, 1.64491124363510222152541210383, 1.91908160474091000860488401732, 2.78150680868566652829432988207, 2.94763947132601989060785544738, 3.59762279093051355611247593360, 3.91341765843496044833779666399, 4.24079695779039540127202302746, 4.92486245114663513451730362960, 5.44634112571062042917450517953, 5.92538701598272688267222747163, 5.98000098094200671993789129484, 6.60572401394442785051130352317, 7.11804143952243880207139070366, 7.12322501217349445116362600144, 7.971484709414468762173790282772, 7.978756701008942807421859585060, 8.558061667358138619338182477038, 8.737746830402323761778979237602, 8.852073757962957248510621342561
