| L(s) = 1 | − 1.84·3-s + 1.66·5-s − 5.08·7-s + 0.394·9-s − 4.67·11-s − 1.66·13-s − 3.07·15-s − 17-s − 6.51·19-s + 9.36·21-s − 2.71·23-s − 2.21·25-s + 4.80·27-s + 1.66·29-s − 7.44·31-s + 8.60·33-s − 8.48·35-s − 9.36·37-s + 3.07·39-s − 6·41-s + 0.658·45-s + 13.2·47-s + 18.8·49-s + 1.84·51-s − 7.69·53-s − 7.80·55-s + 12·57-s + ⋯ |
| L(s) = 1 | − 1.06·3-s + 0.746·5-s − 1.92·7-s + 0.131·9-s − 1.40·11-s − 0.463·13-s − 0.794·15-s − 0.242·17-s − 1.49·19-s + 2.04·21-s − 0.567·23-s − 0.442·25-s + 0.923·27-s + 0.310·29-s − 1.33·31-s + 1.49·33-s − 1.43·35-s − 1.53·37-s + 0.492·39-s − 0.937·41-s + 0.0981·45-s + 1.93·47-s + 2.68·49-s + 0.257·51-s − 1.05·53-s − 1.05·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08453476494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08453476494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.84T + 3T^{2} \) |
| 5 | \( 1 - 1.66T + 5T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 19 | \( 1 + 6.51T + 19T^{2} \) |
| 23 | \( 1 + 2.71T + 23T^{2} \) |
| 29 | \( 1 - 1.66T + 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 + 9.36T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 7.69T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 5.00T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 7.21T + 73T^{2} \) |
| 79 | \( 1 + 7.44T + 79T^{2} \) |
| 83 | \( 1 + 7.36T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.511499700728094978765449647835, −7.35451239873856486411526829176, −6.72888224988072777728914005333, −6.01310117865802403434823542908, −5.69328513186209026730232088089, −4.88542415015083113753156712522, −3.78826543306011391215625692822, −2.81612591920773637757732566832, −2.07372168691640597569569590620, −0.15717216124819489558650268447,
0.15717216124819489558650268447, 2.07372168691640597569569590620, 2.81612591920773637757732566832, 3.78826543306011391215625692822, 4.88542415015083113753156712522, 5.69328513186209026730232088089, 6.01310117865802403434823542908, 6.72888224988072777728914005333, 7.35451239873856486411526829176, 8.511499700728094978765449647835
