| L(s) = 1 | − 6·3-s + 8·5-s + 16·7-s + 27·9-s − 8·11-s − 72·13-s − 48·15-s − 36·17-s + 136·19-s − 96·21-s + 256·23-s + 6·25-s − 108·27-s − 152·29-s − 80·31-s + 48·33-s + 128·35-s − 136·37-s + 432·39-s − 436·41-s − 712·43-s + 216·45-s − 224·47-s − 286·49-s + 216·51-s − 344·53-s − 64·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.715·5-s + 0.863·7-s + 9-s − 0.219·11-s − 1.53·13-s − 0.826·15-s − 0.513·17-s + 1.64·19-s − 0.997·21-s + 2.32·23-s + 0.0479·25-s − 0.769·27-s − 0.973·29-s − 0.463·31-s + 0.253·33-s + 0.618·35-s − 0.604·37-s + 1.77·39-s − 1.66·41-s − 2.52·43-s + 0.715·45-s − 0.695·47-s − 0.833·49-s + 0.593·51-s − 0.891·53-s − 0.156·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.047818063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047818063\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 542 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 8 T - 650 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 72 T + 4858 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 36 T + 6822 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 136 T + 15014 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 256 T + 39886 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 152 T + 37706 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 80 T + 26030 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 136 T + 102602 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 436 T + 102166 T^{2} + 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 712 T + 282422 T^{2} + 712 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 224 T + 179422 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 344 T + 280538 T^{2} + 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 324 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 324 T + p^{3} T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 456 T + 174278 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2048 T + 1763566 T^{2} - 2048 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 660 T + 407702 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 496 T + 323534 T^{2} - 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 776 T + 1131046 T^{2} - 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 532 T + 1467382 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1220 T + 586694 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.04168311112348861698382369727, −9.779448909852847382546060161511, −9.376116940647493922457025597649, −9.152022641719708945015589347488, −8.303183989998309634106053644143, −7.970385320073127502585552743213, −7.39592972643012927178675942850, −7.09610897635939671834875333761, −6.55422001671749528302404139658, −6.35944992436599034656280145784, −5.33154424284670448438037634895, −5.18184927906689454548206041247, −4.93416257826124655930273353864, −4.76127678371367661293449945380, −3.39109107281374371855001263709, −3.37133580911363636842801467903, −2.29645707107620623293386985922, −1.73238364787003067350675067632, −1.25834291926636198745532918757, −0.30873367226111079688215583878,
0.30873367226111079688215583878, 1.25834291926636198745532918757, 1.73238364787003067350675067632, 2.29645707107620623293386985922, 3.37133580911363636842801467903, 3.39109107281374371855001263709, 4.76127678371367661293449945380, 4.93416257826124655930273353864, 5.18184927906689454548206041247, 5.33154424284670448438037634895, 6.35944992436599034656280145784, 6.55422001671749528302404139658, 7.09610897635939671834875333761, 7.39592972643012927178675942850, 7.970385320073127502585552743213, 8.303183989998309634106053644143, 9.152022641719708945015589347488, 9.376116940647493922457025597649, 9.779448909852847382546060161511, 10.04168311112348861698382369727
