L(s) = 1 | − 10·3-s + 26·4-s − 10·5-s − 70·7-s − 87·9-s + 178·11-s − 260·12-s + 10·13-s + 100·15-s + 420·16-s + 970·17-s − 260·20-s + 700·21-s − 525·25-s + 1.93e3·27-s − 1.82e3·28-s + 382·29-s − 1.78e3·33-s + 700·35-s − 2.26e3·36-s − 100·39-s + 4.62e3·44-s + 870·45-s + 4.39e3·47-s − 4.20e3·48-s + 2.49e3·49-s − 9.70e3·51-s + ⋯ |
L(s) = 1 | − 1.11·3-s + 13/8·4-s − 2/5·5-s − 1.42·7-s − 1.07·9-s + 1.47·11-s − 1.80·12-s + 0.0591·13-s + 4/9·15-s + 1.64·16-s + 3.35·17-s − 0.649·20-s + 1.58·21-s − 0.839·25-s + 2.64·27-s − 2.32·28-s + 0.454·29-s − 1.63·33-s + 4/7·35-s − 1.74·36-s − 0.0657·39-s + 2.39·44-s + 0.429·45-s + 1.98·47-s − 1.82·48-s + 1.04·49-s − 3.72·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.395016627\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.395016627\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | $C_2$ | \( 1 + 2 p T + p^{4} T^{2} \) |
| 7 | $C_2$ | \( 1 + 10 p T + p^{4} T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 13 p T^{2} + p^{8} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 5 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 89 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 485 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 212042 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 68906 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 191 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 737642 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 794 p^{2} T^{2} + p^{8} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2845078 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6695306 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2195 T + p^{4} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 13261538 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 11092322 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 23947082 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 36108866 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4454 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8650 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5561 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 1990 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 124831082 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 9235 T + p^{4} T^{2} )^{2} \) |
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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
−16.32059356744224888236700699846, −15.75967767595344737108854696118, −14.98014542926527280332458792812, −14.34350143146305992649782016101, −13.95439614707832904826478242169, −12.48867412441494126891130975416, −12.18780273573968715198874847638, −11.82903793454783615842268947815, −11.44070189882575411020251376976, −10.57242669315692193649474573994, −10.07097693322161259283997131319, −9.242173355310444402337589436805, −8.134074552846417593436736188004, −7.30324564692594695078584080633, −6.57354826322717129890171805563, −5.84053442731215097730335420704, −5.70706770608959593601195736919, −3.57956700653466011907015963814, −2.94758631634964076960185359270, −0.928285038264232944680980413603,
0.928285038264232944680980413603, 2.94758631634964076960185359270, 3.57956700653466011907015963814, 5.70706770608959593601195736919, 5.84053442731215097730335420704, 6.57354826322717129890171805563, 7.30324564692594695078584080633, 8.134074552846417593436736188004, 9.242173355310444402337589436805, 10.07097693322161259283997131319, 10.57242669315692193649474573994, 11.44070189882575411020251376976, 11.82903793454783615842268947815, 12.18780273573968715198874847638, 12.48867412441494126891130975416, 13.95439614707832904826478242169, 14.34350143146305992649782016101, 14.98014542926527280332458792812, 15.75967767595344737108854696118, 16.32059356744224888236700699846